2·1. Reduction modulo prime powers of cyclotomic polynomials

Lemma 2·1. Let $d \geqslant 1$ be an integer and let $q = p^\alpha$ be a d-admissible integer. Let $x \in {{\left({{\textbf{Z}}}/q {{\textbf{Z}}}\right)^\times}}$ be an element of order d. Then we have:

\begin{equation*}\overline \phi_d (x) = 0 \text{ in } {{{{\textbf{Z}}}/q {{\textbf{Z}}}}},\end{equation*}

where $\overline \phi_d$ stands for the reduction of $\phi_d$ modulo q.

Proof. We consider the polynomial $P(X) \,:\!=\, X^{d} -1$ , seen as an element in ${{\textbf{Z}}}_p[X]$ , where ${{\textbf{Z}}}_p$ is the ring of p-adic integers. Let $\tilde x$ be a lift in ${{\textbf{Z}}}$ of the class x modulo q. Then we have

\begin{equation*}P(\tilde x) \equiv 0 {\,\textrm{mod}\,} q\end{equation*}

since x has order d. Therefore $ |P(\tilde x) |_p \leqslant {1}/{p^\alpha}$ , where we denoted by $|\cdot |_p$ the standard p-adic absolute value on the field of p-adic numbers ${{\textbf{Q}}}_p$ . On the other hand, we have $P^{\prime}(\tilde x) = d\tilde x^{d-1}$ , which has p-adic valuation zero since $(d,p) = 1$ (because d divides $p-1$ ) and $(\tilde x,p)= 1$ since x is invertible modulo $p^\alpha$ . Thus, $|P^{\prime}(\tilde x) |_p = 1$ and so:

\begin{equation*} |P(\tilde x) |_p \leqslant \frac{1}{p^\alpha} =\frac{1}{p^\alpha} |P^{\prime}(\tilde x) |^2_p.\end{equation*}

Therefore, by Hensel’s lemma (see [ Reference Cassels and Fröhlich5 , chapter II, appendix C]) there exists a unique $ \tilde z\in {{\textbf{Z}}}_p$ such that

(2·1) \begin{equation} \begin{cases} P(\tilde z) = 0 \\ |\tilde z -\tilde x|_p \leqslant \frac{1}{p^\alpha}. \end{cases} \end{equation}

We deduce that:

(2·2) \begin{equation} 0 = \tilde z^d - 1 = \prod_{m \mid d} \phi_m(\tilde z) \quad \text{in } {{\textbf{Z}}}_p. \end{equation}

Now since ${{\textbf{Z}}}_p$ is an integral domain, at least one of the factors $\phi_m( \tilde z)$ must be zero.

Assume for a contradiction that this happens for an m which is not equal to d. Then this would imply that $\tilde z^m = 1 $ in ${{\textbf{Z}}}_p$ , hence:

\begin{equation*} |\tilde x^m - 1|_p =|\tilde x^m - \tilde z^m |_p \leqslant |\tilde x - \tilde z|_p \leqslant \frac{1}{p^\alpha}\end{equation*}

by the second condition in (2·1). Thus, $\tilde x^m \equiv 1 {\,\textrm{mod}\,} p^\alpha$ for an $m < d$ , contradicting the fact that x has order exactly d in $ {{\left({{\textbf{Z}}}/p^\alpha {{\textbf{Z}}}\right)^\times}}$ . Therefore, in the product (2·2), it is the term $\phi_d(\tilde z)$ which equals zero. Now, since $|\tilde x - \tilde z|_p \leqslant {1}/{p^\alpha}$ we have:

\begin{equation*}|\phi_d(\tilde x) |_p =|\phi_d( \tilde x) - \phi_d(\tilde z) |_p \leqslant \frac{1}{p^\alpha}\end{equation*}

and this is equivalent to $\phi_d(\tilde x) \equiv 0 {\,\textrm{mod}\,} p^\alpha$ , that is: $\overline \phi_d(x) = 0$ in ${{\textbf{Z}}}/p^\alpha {{\textbf{Z}}}$ .

In the remainder of this section, we state two lemmas on the equidistribution modulo 1 of some particular sets, and prove that they imply Theorem A.

2·2. Reduction step for the main result: case (a)

First, let us state the proposition which will turn out to imply case (a) of Theorem A.

Proposition 2·2. (Equidistribution modulo 1 case (a)) Let $d \geqslant 1$ be an integer and ${{\textbf{m}}} =(m_1, \dots, m_n) \in {{\textbf{Z}}}^n$ . For all $i \in \{1, \dots, n\}$ , the notation $d_i$ stands for ${d}/{(d,m_i)}$ . Let $\delta > 0$ . For all d-admissible integers q, let $w_q$ be an element of order d in ${{\left({{\textbf{Z}}}/q {{\textbf{Z}}}\right)^\times}}$ . For each such q, we also choose subgroups of ${{\left({{\textbf{Z}}}/q {{\textbf{Z}}}\right)^\times}}$ : $H^{(1)}_q , \dots , H^{(n)}_q$ . Then, provided the subgroups satisfy the growth conditions:

\begin{equation*}\forall i \in \{1, \dots,n\}, \qquad |H_{q}^{(i)}| \geqslant q^\delta, \end{equation*}

the sets of $(\varphi(d_1) + \dots + \varphi(d_n))$ -tuples

\begin{equation*}\bigg \{ \overbrace{\left( \left(\frac{a_1 ( w_q^{m_1})^j}{q}\right)_{0 \leqslant j < \varphi(d_1)} , \dots, \left(\frac{a_n (w_q^{m_n})^j}{q}\right)_{0 \leqslant j < \varphi(d_n)} \right)}^{=: {{\textbf{x}}}(a_1, \dots, a_n,q)}; \quad (a_1, \dots, a_n) \in H^{(1)}_q \times \dots \times H^{(n)}_q \bigg \} \end{equation*}

become equidistributed modulo 1 as q goes to infinity among the d-admissible integers. In other words, for any continuous map $G \,:\, [0,1]^d \to {{\textbf{C}}}$ ,

\begin{equation*} \frac{1}{\prod_{i = 1}^n |H_q^{(i)}|} \sum_{\substack{a_1 \in H^{(1)}_q \\ \vdots \\ a_n \in H_q^{(n)}}} G(\{ {{\textbf{x}}}(a_1, \dots, a_n,q)\}) \underset{\substack{q \to \infty \\ q \in {\mathcal A}_d}}{\longrightarrow} \int_{[0,1]^d} G(x_1, \dots, x_d) {{\textrm{d}}} x_1 \dots {{\textrm{d}}} x_d.\end{equation*}

Proof. See Section 4.1.

This lemma on equidistribution modulo 1 translates into a result on equidistribution of exponential sums restricted to a subgroup via the Laurent polynomials $f_{d, {{\textbf{m}}}}$ introduced in equation (1·7) of Definition 1·9.

Proof. Let $d\geqslant 1$ and ${{\textbf{m}}} = (m_1, \dots, m_n) \in {{\textbf{Z}}}^n$ . Let us denote by $\mathcal I_{d, {{\textbf{m}}}}$ the image of ${{\mathbb{T}}}^{\varphi(d_1) +\dots + \varphi(d_n)}$ via $f_{d, {{\textbf{m}}}}$ and by $\mu \,:\!=\, (f_{d, {{\textbf{m}}}})_*\lambda$ the pushforward measure of the probability Haar measure $\lambda$ on ${{\mathbb{T}}}^{\varphi(d_1) +\dots + \varphi(d_n)}$ .

For all d-admissible integers q, let

\begin{equation*}Y_{{{\textbf{m}}},q} \,:\!=\, H^{(1)}_{q} \times \dots \times H^{(n)}_{q}\end{equation*}

be a product of subgroups of ${{\left({{\textbf{Z}}}/q {{\textbf{Z}}}\right)^\times}}$ as in case (a) of Theorem A (that is: satisfying the growth condition (1·8)) and let $\theta_{{{\textbf{m}}}, q} \,:\, Y_{{{\textbf{m}}},q} \to {{\textbf{C}}} $ be the map defined by

(2·3) \begin{equation} (a_1, \dots, a_n) \longmapsto \sum_{\substack{x \in ({{{{\textbf{Z}}}/q {{\textbf{Z}}}}})^\times \\ x^d = 1}} e\left(\frac{a_1 x^{m_1} + \dots + a_n x^{m_n}}{q}\right). \end{equation}

We want to show that, assuming Proposition 2·2, we have

(2·4) \begin{equation} \frac{1}{|Y_{{{\textbf{m}}},q}|} \sum_{(a_1, \dots ,a_n) \in Y_{{{\textbf{m}}},q}} F\left(\theta_{{{\textbf{m}}},q} \left( a_1, \dots, a_n\right) \right) \underset{\substack{q \to \infty \\ q \in {\mathcal A}_d}}{\longrightarrow} \int_{\mathcal I_{d, {{\textbf{m}}}}}^{} F {{\textrm{d}}} \mu, \end{equation}

for any $F\,:\, \mathcal I_{d, {{\textbf{m}}}} \to {{\textbf{C}}}$ continuous.

Let $w_q$ be an element of order d in $({{{{\textbf{Z}}}/q {{\textbf{Z}}}}})^\times$ . Then the unique subgroup of order d inside ${{\left({{\textbf{Z}}}/q {{\textbf{Z}}}\right)^\times}}$ can be described as $\left\{ w_q^k; \ k \in \{0, \dots, d-1\}\right\}$ , so that for all $(a_1, \dots, a_n) \in Y_{ {{\textbf{m}}},q},$

\begin{equation*}\theta_{{{\textbf{m}}},q}\left( a_1, \dots, a_n\right) =\sum ^{d-1}_{k=0} \prod_{i=1}^{n} e\left(\frac{a_i (w_q^{m_i})^k}{q}\right).\end{equation*}

Now,

\begin{equation*} \forall i \in \{1, \dots, n\},\ \forall k \in \{0,\dots,d-1\}, \qquad (w_q^{m_i})^k = \sum_{j= 0}^{\varphi(d_i) - 1} c_{j,k}^{(i)} (w_q^{m_i})^j.\end{equation*}

This comes from using Lemma 2·1 after having evaluated the polynomial congruences defining the $c_{j,k}^{(i)}$ (see Definition 1·9) at $w_q^{m_i}$ . The lemma applies since $w_q^{m_i}$ has order $d_i$ in ${{\left({{\textbf{Z}}}/q {{\textbf{Z}}}\right)^\times}}$ . Replacing this in the expression of $\theta_{{{\textbf{m}}},q}\left( a_1, \dots, a_n\right)$ obtained above, we get:

\begin{equation*}\theta_{{{\textbf{m}}},q}\left( a_1, \dots, a_n\right) = \sum ^{d-1}_{k=0} \prod_{i=1}^{n} \prod_{j=0}^{\varphi(d_i)-1} e\left(\frac{a_i (w_q^{m_i})^j}{q}\right)^{c_{j,k}^{(i)}}.\end{equation*}

Therefore, if we define for all $i \in \{1, \dots, n\}$ and for all $j \in \{ 0, \dots, \varphi(d_i)-1 \}$ ,

\begin{equation*} z_{i,j} = z_{i,j}(a_1, \dots, a_n, q) \,:\!=\, e\left(\frac{a_i (w_q^{m_i})^j}{q}\right) \end{equation*}

we have

(2·5) \begin{equation} \theta_{{{\textbf{m}}},q} \left( a_1, \dots, a_n\right) = f_{d, {{\textbf{m}}}} \left((z_{1,j})_{0 \leqslant j < \varphi(d_1)}, \dots, (z_{n,j})_{0 \leqslant j< \varphi(d_n)}\right) \end{equation}

with the Laurent polynomial $f_{d, {{\textbf{m}}}}$ from Definition 1·9, and the $z_{i,j}$ being elements of ${{\mathbb{T}}}$ . This already shows that $\theta_{{{\textbf{m}}},q} \left( a_1, \dots, a_n\right) $ belongs to the image of $f_{d, {{\textbf{m}}}}$ .

Now, let us prove that the equidistribution statement. Let $F\,:\, \mathcal I_{d, {{\textbf{m}}}} \to {{\textbf{C}}}$ be a continuous function. Thanks to (2·5), the left-hand side of (2·4) may be rewritten as

\begin{equation*}\frac{1}{|Y_{{{\textbf{m}}},q}|} \sum_{(a_1, \dots, a_n) \in Y_{{{\textbf{m}}},q}} F\left( f_{d, {{\textbf{m}}}}\left((z_{1,j})_{0 \leqslant j < \varphi(d_1)}, \dots, (z_{n,j})_{0 \leqslant j < \varphi(d_n)}\right) \right)\end{equation*}

and this converges to

\begin{equation*} \int_{{{\mathbb{T}}}^{\varphi(d_1) +\dots + \varphi(d_n)}}^{}\left( F \circ f_{d, {{\textbf{m}}}} \right) {{\textrm{d}}} \lambda = \int_{\mathcal I_{d, {{\textbf{m}}}}}^{} F {{\textrm{d}}} \mu\end{equation*}

by Proposition 2·2 and because $\mu = (f_{d, {{\textbf{m}}}})_*\lambda$ . This finishes the proof of (2·4). Thus, Theorem A (a) indeed follows from Proposition 2·2.

2·3. Reduction step for the main result: case (b)

As in the previous case, let us begin with the statement of the proposition which will imply case (b) of Theorem A.

Proposition 2·5 (Equidistribution modulo 1 case (b)). Let $d \geqslant 1$ be an integer and ${{\textbf{m}}} =(m_1, \dots, m_n) \in {{\textbf{Z}}}^n$ be a vector coprime with d. Let $\delta > 0$ . For all d-admissible integers q, let $w_q$ be an element of order d in ${{\left({{\textbf{Z}}}/q {{\textbf{Z}}}\right)^\times}}$ . For each q, we also choose subgroups of ${{\left({{\textbf{Z}}}/q {{\textbf{Z}}}\right)^\times}}$ : $H^{(1)}_q , \dots , H^{(n)}_q$ . Then, provided for all q, there exists $i \in \{1, \dots, n\}$ such that

\begin{equation*}|H_{q}^{(i)}| \geqslant q^\delta, \end{equation*}

the sets of $\varphi(d)$ -tuples

\begin{multline*} \bigg\{ \left(\frac{a_1 (w_q^{m_1})^0 + \dots + a_n (w_q^{m_n})^0}{q}, \dots, \frac{a_1 (w_q^{m_1})^{\varphi(d)-1} + \dots + a_n (w_q^{m_n})^{\varphi(d)-1}}{q} \right); \\ (a_1, \dots, a_n) \in H^{(1)}_q \times \dots \times H^{(n)}_q \bigg \} \end{multline*}

become equidistributed modulo 1 as q goes to infinity among the d-admissible integers.

Proof. See Section 4·2.

Corollary 1·14, which generalises the classical Myerson’s lemma stated in the introduction, is obtained as the special case where ${{\textbf{m}}} = (1)$ .

The Laurent polynomials $g_d$ introduced in Definition 1·2 will carry the equidistribution result of Proposition 2·5 to the equidistribution theorem for exponential sums we are aiming at. This is the content of the following proposition.

Proof. We still denote by $Y_{{{\textbf{m}}},q} \,:\!=\, H_q^{(1)} \times \dots \times H_q^{(n)}$ and by $\theta_{{{\textbf{m}}}, q}$ the map defined in (2·3). We assume that the subgroups $H_q^{(i)}$ satisfy the growth condition (1·9) instead of (1·8). Thanks to the proof of Proposition 2·4 we have that

\begin{equation*}\theta_{{{\textbf{m}}},q}\left( a_1, \dots, a_n\right) = \sum ^{d-1}_{k=0} \prod_{i=1}^{n} \prod_{j=0}^{\varphi(d_i)-1} e\left(\frac{a_i (w_q^{m_i})^j}{q}\right)^{c_{j,k}^{(i)}} \end{equation*}

for all $(a_1, \dots, a_n) \in Y_{ {{\textbf{m}}},q}$ , where the integers $c_{j,k}^{(i)}$ are defined as in Definition 1·9. However, in the case where ${{\textbf{m}}}$ is coprime with d, all the $d_i$ are equal to d. This implies that for all i, the list of integers $(c_{j,k}^{(i)})_{0 \leqslant j <\varphi(d_i), \ 0 \leqslant k < d}$ is always equal to the list $(c_{j,k})_{0 \leqslant j <\varphi(d), \ 0 \leqslant k < d}$ of Definition 1·2. Therefore,

\begin{align*}\theta_{{{\textbf{m}}},q}\left( a_1, \dots, a_n\right)= \sum ^{d-1}_{k=0} \prod_{j=0}^{\varphi(d)-1} e\left(\frac{a_1 (w_q^{m_1})^j + \dots + a_n (w_q^{m_n})^j}{q}\right)^{c_{j,k}}. \end{align*}

So if we define for all $j \in \{ 0, \dots, \varphi(d)-1 \}$ ,

(2·6) \begin{equation} z_j = z_j(a_1, \dots, a_n, q, j) \,:\!=\, e\left(\frac{a_1 (w_q^{m_1})^j + \dots + a_n (w_q^{m_n})^j}{q}\right), \end{equation}

we have

(2·7) \begin{equation} \theta_{{{\textbf{m}}},q} \left( a_1, \dots, a_n\right) = g_d(z_0, \dots, z_{\varphi(d)-1}), \end{equation}

with the Laurent polynomial $g_d$ defined at Definition 1·2 and the $z_j$ being elements of ${{\mathbb{T}}}$ . This shows that $\theta_{{{\textbf{m}}},q} \left( a_1, \dots, a_n\right) $ belongs to the image of $g_d$ .

Now, let us prove that the sums $\theta_{{{\textbf{m}}},q} \left( a_1, \dots, a_n\right)$ become equidistributed in the image of $g_d$ with respect to the pushforward measure of the probability Haar measure $\lambda$ on ${{\mathbb{T}}}^{\varphi(d)}$ . We denote by $\mathcal I_d \,:\!=\, g_d\left({{\mathbb{T}}}^{\varphi(d)}\right)$ the image of $g_d$ and by $\mu \,:\!=\, (g_d)_*\lambda$ the pushforward measure. Then what we have to prove is that for any continuous function $F\,:\, \mathcal I_d \to {{\textbf{C}}}$ , we have:

\begin{equation*} \frac{1}{\prod_{i= 1}^{n}|H_{q}^{(i)}|} \sum_ {a_1\in H_{q}^{(1)}} \cdots \sum_{a_n \in H_q^{(n)}} F\left(\theta_{{{\textbf{m}}},q} \left( a_1, \dots, a_n\right) \right) \underset{\substack{ q \to \infty \\ q \in {\mathcal A}_d}}{\longrightarrow} \int_{\mathcal I_{d}}^{} F {{\textrm{d}}} \mu.\end{equation*}

As in the proof of Proposition 2·4, we first use (2·7) to write:

\begin{align*} & \frac{1}{\prod_{i= 1}^{n}|H_{q}^{(i)}|} \sum_ {a_1\in H_{q}^{(1)}} \cdots \sum_{a_n \in H_q^{(n)}} F\left(\theta_{{{\textbf{m}}},q} \left( a_1, \dots, a_n\right) \right) \\ & \quad = \frac{1}{\prod_{i= 1}^{n}|H_{q}^{(i)}|} \sum_ {a_1\in H_{q}^{(1)}} \cdots \sum_{a_n \in H_q^{(n)}} F\left( g_{d}\left(z_0, \dots, z_{\varphi(d)-1}\right) \right) \end{align*}

with the notation $z_j$ from (2·6) (let us stress that the $z_j$ depend on $a_1, \dots, a_n$ ). Since $F \circ g_d$ is continuous and the tuples $\left(z_0, \dots, z_{\varphi(d)-1}\right)$ become equidistributed modulo 1 (thanks to Proposition 2·5), we obtain that

\begin{equation*}\frac{1}{\prod_{i= 1}^{n}|H_{q}^{(i)}|} \sum_ {a_1\in H_{q}^{(1)}} \cdots \sum_{a_n \in H_q^{(n)}} F\left( g_{d}\left(z_0, \dots, z_{\varphi(d)-1}\right) \right) \underset{\substack{q \to \infty \\ q \in {\mathcal A}_d}}{\longrightarrow} \int_{{{\mathbb{T}}}^{\varphi(d)}} (F \circ g_d) {{\textrm{d}}} \lambda.\end{equation*}

Finally, the integral on the righ-hand side equals $\int_{\mathcal I_{d}}^{} F {{\textrm{d}}} \mu $ by definition of the pushforward measure, and this concludes the proof.

In the next section, we prove the key exponential sum estimate which will allow us to prove Proposition 2·2 and Proposition 2·5 using Weyl’s equidistribution criterion.

3·1. Bourgain’s theorem

In a series of articles, Bourgain, Chang, Glibichuk and Konyagin proved very strong estimates on sums of additive characters modulo q over subgroups of ${{\left({{\textbf{Z}}}/q {{\textbf{Z}}}\right)^\times}}$ , for different forms of factorisation of q. The case where q is prime is proved in [ Reference Bourgain, Glibichuk and Konyagin3 ], while the case of prime powers with bounded exponent is settled in [ Reference Bourgain and Chang2 ]. This series of works culminated with the following theorem, which treats the general case, and includes in particular the case of small primes raised to very high powers. Notice that since we are only dealing with moduli which are prime powers, the proof of the result we really use is contained in part I of [ Reference Bourgain1 ].

Theorem 3·1 (Bourgain). For any $\delta > 0$ , there exists $\varepsilon = \varepsilon (\delta) > 0$ such that for any integer $q \geqslant 2$ , and any subgroup H of ${{\left({{\textbf{Z}}}/q {{\textbf{Z}}}\right)^\times}}$ such that $|H| \geqslant q^\delta$ ,

(3·4) \begin{equation} \max_{a\in {{\left({{\textbf{Z}}}/q {{\textbf{Z}}}\right)^\times}}} \left| \sum_{x \in H}^{} e \left( \frac{ax}{q}\right)\right| \leqslant C \frac{|H|}{q^\varepsilon}, \end{equation}

where C is a constant depending at most on $\delta$ .

Proof. See [ Reference Bourgain1 , theorem].

3·2. The crucial control of the p-adic valuation

In order to apply the previous theorem in our specific context, the following proposition plays an important role.

Proof. As in the proof of [ Reference Duke, Garcia and Lutz7 , lemma 6·2], we use the fact that there exist two polynomials $a,b \in {{\textbf{Z}}}[X]$ and an integer $n \geqslant 1$ such that

(3·5) \begin{equation} a(X)\phi_d(X) + b(X) f(X) = n, \end{equation}

since f and $\phi_d$ are coprime in the euclidean domain ${{\textbf{Q}}}[X]$ . Now, let $q= p^\alpha$ be a d-admissible integer. Reducing equation (3·5) modulo q and evaluating it at $w_q$ leads to

\begin{equation*} \overline a ( w_q) \overline \phi_d( w_q) + \overline b ( w_q) \overline f ( w_q) \equiv n {\,\textrm{mod}\,} p^\alpha \end{equation*}

hence

(3·6) \begin{equation} \overline b ( w_q) \overline f ( w_q) \equiv n {\,\textrm{mod}\,} p^\alpha \end{equation}

by Lemma 2·1. Now, if $q = p^\alpha > n$ , then n is non-zero modulo q, hence $\overline f ( w_q) \not \equiv 0 {\,\textrm{mod}\,} q$ . This precisely means that q does not divide $f(\tilde w_q)$ . This shows that $n_f\,:\!=\, n$ is a suitable constant for assertion (a). This part of the lemma actually completes the proof of Lemma 1·13.

Another way of phrasing what we just proved is that as soon as $q > n$ , the p-adic valuation of $f(\tilde w_q)$ is strictly less than $\alpha$ . Let us denote by $\gamma < \alpha$ the p-adic valuation of $f(\tilde w_q)$ . Then, if we reduce the congruence (3·6) modulo $p^\gamma$ , we get $n \equiv 0 {\,\textrm{mod}\,} p^\gamma$ . Thus,

\begin{equation*}\gamma = v_p(f(\tilde w_q)) \leqslant v_p(n),\end{equation*}

hence

\begin{equation*}p^{v_p(f(\tilde w_q))} \leqslant p^{ v_p(n)} \leqslant n.\end{equation*}

Therefore, we proved that with the choice $C_f \,:\!=\, n$ , assertion (b) holds.

Now, let us use this proposition, together with Theorem 3·1, to deduce the key exponential sum estimate which will be used in the proofs of Proposition 2·2 and Proposition 2·5.

3·3. Proof of the key exponential sum estimate: Proposition 1·15

Let $q = p^ \alpha$ be a d-admissible integer, and let $H_q$ and $w_q$ be as in the statement. Let $ \tilde w_q$ be any lift in $ {{\textbf{Z}}}$ of the class $w_q$ . Assume further that $q > n_f$ for any constant $n_f$ as in Proposition 3·2.

One cannot directly apply Bourgain’s theorem to the sum on the left-hand side of (1·13) because $f(w_q)$ could be non-invertible modulo q. In order to reduce to a situation where Bourgain’s theorem applies, let us introduce the notation $ \beta_q$ for the p-adic valuation of $f(\tilde w_q)$ , and write $f(\tilde w_q) \,:\!=\, p^{\beta_q} r_q$ . By Proposition 3·2 (a), $\beta_q < \alpha$ . Then we have

(3·7) \begin{equation} \sum_{a \in H_{q}}^{} e \left( \frac{af(w_{q})}{q}\right) = \sum_{a \in H_{q}}^{} e \left( \frac{ar_q}{p^{\alpha - \beta_q}}\right).\end{equation}

Now, each of the terms $e \left( {ar_q}/{p^{\alpha - \beta_q}}\right)$ only depends on the class of a modulo $p^{\alpha - \beta_q}$ . Let us denote by $q^{\prime} \,:\!=\, p^{\alpha - \beta_q}$ and by $\pi$ the group homomorphism $({{\textbf{Z}}}/ q {{\textbf{Z}}})^\times \to ({{\textbf{Z}}} / q^{\prime} {{\textbf{Z}}})^\times$ induced by the reduction modulo q’. The latter induces a group homomorphism $\tilde \pi \,:\, H_q \to \pi(H_q) =: H^{\prime}_q$ . We denote by $k \,:\!=\, |\ker \tilde \pi |$ . Then we have the following equality:

\begin{equation*}\sum_{a \in H_{q}}^{} e \left( \frac{ar_q}{p^{\alpha - \beta_q}}\right) = k\sum_{a \in H^{\prime}_{q}}^{} e \left( \frac{ar_q}{q^{\prime}}\right).\end{equation*}

Indeed, any element of $H^{\prime}_q$ has exactly k preimages in $H_q$ under the reduction modulo q’.

Since $\ker \tilde \pi \subseteq \ker \pi$ , we have that $k \leqslant | \ker \pi | = p^{\beta_q}$ . Therefore,

(3·8) \begin{equation} \left|\sum_{a \in H_{q}}^{} e \left( \frac{ar_q}{p^{\alpha - \beta_q}}\right)\right| \leqslant p^{\beta_q} \left| \sum_{a \in H^{\prime}_{q}}^{} e \left( \frac{ar_q}{q^{\prime}}\right) \right|.\end{equation}

In order to apply Theorem 3·1 to the sum on the right-hand side, we first need to check that the subgroup $H^{\prime}_q$ of $\left( {{\textbf{Z}}} / q^{\prime} {{\textbf{Z}}} \right)^\times$ is large in the following sense: $|H^{\prime}_q| \geqslant (q^{\prime})^{\delta^{\prime}}$ for some $\delta^{\prime} > 0$ . Using the fact that $|H_q| \geqslant q^\delta$ , we have

\begin{equation*} |H^{\prime}_q| = \frac{|H_q|}{k} \geqslant \frac{|H_q|}{p^{\beta_q}} \geqslant \frac{q^\delta}{p^{\beta_q}} = \frac{(q^{\prime})^\delta}{\left(p^{\beta_q}\right)^{1-\delta}} \geqslant \frac{(q^{\prime})^\delta}{C_f^{1-\delta}},\end{equation*}

where the last lower bound comes from the inequality $p^{\beta_q} \leqslant C_f$ given by Proposition 3·2 (b). This inequality also ensures that q’ tends to infinity as q goes to infinity, so that ${(q^{\prime})^{\frac{\delta}{2}}}/{C_f^{1- \delta}}$ eventually becomes greater than 1 as q becomes large. Therefore,

(3·9) \begin{equation} |H^{\prime}_q| \geqslant (q^{\prime})^{\frac{\delta}{2}}\end{equation}

for all q large enough. Thanks to (3·9), Theorem 3·1 applies to the sum on the right-hand side of (3·8), because we also have that $r_q$ is invertible modulo q’. So there exists a constant $\varepsilon = \varepsilon(\delta /2) > 0$ and a constant C depending at most on $\delta$ such that

\begin{equation*}\left| \sum_{a \in H^{\prime}_{q}}^{} e \left( \frac{ar_q}{q^{\prime}}\right) \right| \leqslant C \frac{|H^{\prime}_q|}{(q^{\prime})^\varepsilon} \cdot\end{equation*}

Thanks to (3·7) and (3·8), this implies the following upper bound:

\begin{equation*}\left|\sum_{a \in H_{q}}^{} e \left( \frac{af(w_{q})}{q}\right)\right| \leqslant C \frac{p^{\beta_q}|H^{\prime}_q|}{(q^{\prime})^\varepsilon} \leqslant C \frac{p^{\beta_q}|H_q|}{\left(p^{\alpha - \beta_q}\right)^\varepsilon} = C \frac{|H_q|p^{\beta_q (1+\varepsilon)}}{q^\varepsilon} \leqslant CC_f^{1+\varepsilon} \frac{|H_q|}{q^\varepsilon} \ll_{f, \delta} \frac{|H_q|}{q^\varepsilon}\end{equation*}

which concludes the proof.

4·1. Proof of Proposition 2·2 (Equidistribution modulo 1 case (a))

We are interested in the equidistribution modulo 1 of the following sets of $(\varphi(d_1) + \dots + \varphi(d_n))$ -tuples (with the slight abuse of notation underlined at Remark 2·3):

\begin{equation*}\bigg \{ \overbrace{\left( \left(\frac{a_1 ( w_q^{m_1})^j}{q}\right)_{0 \leqslant j < \varphi(d_1)} , \dots, \left(\frac{a_n (w_q^{m_n})^j}{q}\right)_{0 \leqslant j < \varphi(d_n)} \right)}^{=: {{\textbf{x}}}(a_1, \dots, a_n,q)}; \quad (a_1, \dots, a_n) \in H^{(1)}_q \times \dots \times H^{(n)}_q \bigg \},\end{equation*}

where the $H_q^{(i)}$ are subgroups of ${{\left({{\textbf{Z}}}/q {{\textbf{Z}}}\right)^\times}}$ satisfying the following growth condition:

\begin{equation*}\forall q \in {\mathcal A}_d, \forall i \in \{1, \dots, n\}, \ |H_q^{(i)}| \geqslant q^\delta\end{equation*}

for some $\delta > 0$ . By Weyl’s criterion (see [ Reference Kuipers and Niederreiter14 , theorem 6·2]), these sets become equidistributed modulo 1 if and only if for any ${{\textbf{y}}} = \left(y_0, \dots, y_{\varphi(d_1)+ \dots + \varphi(d_n)-1}\right) \in {{\textbf{Z}}}^{\varphi(d_1)+ \dots + \varphi(d_n)} \setminus \{0\}$ , we have the following convergence towards zero:

(4·1) \begin{equation} \frac{1}{\prod_{i= 1}^{n} |H_q^{(i)}|} \times \ \sum_{\substack{a_1 \in H^{(1)}_q \\ \vdots \\ a_n \in H_q^{(n)}}} e\left({{\textbf{x}}}(a_1,\dots,a_n,q)\cdot {{\textbf{y}}}\right) \underset{\substack{q \to \infty \\ q \in {\mathcal A}_d}}{\longrightarrow} 0.\end{equation}

Let us denote by ${{\textbf{y}}}_1$ the vector extracted from ${{\textbf{y}}}$ by taking the first $\varphi(d_1)$ entries, ${{\textbf{y}}}_2$ the vector formed by the next $\varphi(d_2)$ entries and so on:

\begin{equation*}{{\textbf{y}}}_1 = (y_0, \dots, y_{\varphi(d_1)-1}), \quad {{\textbf{y}}}_2 = (y_{\varphi(d_1)} , \dots, y_{\varphi(d_1) + \varphi(d_2) - 1}), \quad {{\textbf{y}}}_3 = \cdots \end{equation*}

so that ${{\textbf{y}}} = ({{\textbf{y}}}_1, \dots, {{\textbf{y}}}_n)$ . We also introduce the following notations to decompose the vector ${{\textbf{x}}}(a_1, \dots, a_n,q)$ in a parallel way:

\begin{equation*}{{\textbf{x}}}_1(a_1,q) \,:\!=\, \left(\frac{a_1 ( w_q^{m_1})^j}{q}\right)_{0 \leqslant j < \varphi(d_1)}, \dots,\ {{\textbf{x}}}_n(a_n,q)\,:\!=\, \left(\frac{a_n ( w_q^{m_n})^j}{q}\right)_{0 \leqslant j < \varphi(d_n)}.\end{equation*}

Then we have

(4·2) \begin{equation} \frac{1}{\prod_{i= 1}^{n} |H_q^{(i)}|} \times \ \sum_{\substack{a_1 \in H^{(1)}_q \\ \vdots \\ a_n \in H_q^{(n)}}} e\left({{\textbf{x}}}(a_1,\dots,a_n,q)\cdot {{\textbf{y}}}\right) =\prod_{i=1}^{n} \left[\frac{1}{ |H_q^{(i)}|} \sum_{a_i \in H_q^{(i)}}^{} e({{\textbf{x}}}_i(a_i,q) \cdot {{\textbf{y}}}_i)\right].\end{equation}

Now, since ${{\textbf{y}}} \neq 0$ , there exists at least one index $i \in \{1, \dots, n\}$ such that ${{\textbf{y}}}_{i} \neq 0$ . For such an i, the factor

(4·3) \begin{equation} \frac{1}{ |H_q^{(i)}|} \sum_{a_i \in H_q^{(i)}}^{} e({{\textbf{x}}}_i(a_i,q) \cdot {{\textbf{y}}}_i)\end{equation}

tends to 0 as q goes to infinity among the d-admissible integers, thanks to Proposition 1·15. Indeed, we have

\begin{equation*}\frac{1}{ |H_q^{(i)}|} \sum_{a_i \in H_q^{(i)}}^{} e({{\textbf{x}}}_i(a_i,q) \cdot {{\textbf{y}}}_i)= \frac{1}{|H_q^{(i)}|} \sum_{a \in H_q^{(i)}}^{} e\left( \frac{af_i(w_q^{m_i})}{q} \right),\end{equation*}

where $f_i$ is the polynomial associated with

\begin{equation*}{{\textbf{y}}}_i =\left(y_{\varphi(d_1) + \dots + \varphi(d_{i-1})}, \dots, y_{\varphi(d_1) + \dots + \varphi(d_{i-1}) + \varphi(d_i)-1} \right)\end{equation*}

as follows:

\begin{equation*}f_i = y_{\varphi(d_1) + \dots + \varphi(d_{i-1})} + y_{\varphi(d_1) + \dots + \varphi(d_{i-1}) +1} X + \dots + y_{\varphi(d_1) + \dots +\varphi(d_{i-1}) + \varphi(d_i)-1} X^{\varphi(d_i)-1} .\end{equation*}

This is a non-zero polynomial with integer coefficients and with degree strictly less than $\varphi(d_i)$ , and $w_q^{m_i}$ is an element of order $d_i$ in ${{\left({{\textbf{Z}}}/q {{\textbf{Z}}}\right)^\times}}$ . Thus, we can apply Proposition 1·15 which states that there exists a rank $N_{f_i}$ such that for all $q > N_{f_i}$ such that q is d-admissible,

\begin{equation*} \left| \sum_{a \in H_q^{(i)}}^{} e\left( \frac{af_i( w_q^{m_i})}{q} \right)\right| \ll_{\delta, f_i} \frac{|H_q^{(i)}|}{q^{\varepsilon(\delta)}}, \end{equation*}

and this suffices to prove the convergence of (4·3) towards zero. As all the other factors of (4·2) have absolute value bounded above by 1, the whole product converges to zero, and this concludes the proof.

4·2. Proof of Proposition 2·4 (Equidistribution modulo 1 case (b))

We are interested in the equidistribution modulo 1 of the following sets of $\varphi(d)$ -tuples:

\begin{multline*} \bigg\{ \overbrace{\left(\frac{a_1 ( w_q^{m_1})^0 + \dots + a_n ( w_q^{m_n})^0}{q}, \dots, \frac{a_1 ( w_q^{m_1})^{\varphi(d)-1} + \dots + a_n ( w_q^{m_n})^{\varphi(d)-1}}{q} \right)}^{=: {{\textbf{x}}}(a_1,\dots, a_n, q)}; \\ (a_1, \dots, a_n) \in H^{(1)}_q \times \dots \times H^{(n)}_q \bigg \},\end{multline*}

where the $H_q^{(i)}$ are subgroups of ${{\left({{\textbf{Z}}}/q {{\textbf{Z}}}\right)^\times}}$ satisfying the growth condition

(4·4) \begin{equation} \forall q \in {\mathcal A}_d, \exists i \in \{1, \dots, n\}, \quad |H_q^{(i)}| \geqslant q^{\delta}.\end{equation}

By Weyl’s criterion, these sets become equidistributed if and only if for any ${{\textbf{y}}} \,:\!=\, \left(y_0, \dots, y_{\varphi(d) -1} \right) \in {{\textbf{Z}}}^{\varphi(d)} \setminus \{0\}$ we have the following convergence towards zero:

(4·5) \begin{equation} \frac{1}{\prod_{i= 1}^{n} |H_q^{(i)}|}\times \ \left( \sum_{(a_1, \dots, a_n) \in H^{(1)}_q \times \dots \times H^{(n)}_q}^{} e\left({{\textbf{x}}}(a_1,\dots,a_n,q)\cdot {{\textbf{y}}}\right) \right) \underset{\substack{q \to \infty \\ q \in {\mathcal A}_d}}{\longrightarrow} 0\end{equation}

But the left-hand side can be rewritten as:

(4·6) \begin{equation} \prod_{i= 1}^{n} \left[ \frac{1}{|H_q^{(i)}|} \sum_{a_i \in H_q^{(i)}}^{} e\left( \frac{a_if( w_q^{m_i})}{q} \right) \right],\end{equation}

where f is the polynomial $y_0 + y_1X + \dots + y_{\varphi(d)-1} X^{\varphi(d)-1}$ .

Now, since $(m_i,d) = 1$ , we have that for all $i \in \{1, \dots, n\}$ the element $w_q^{m_i}$ is still of order d in ${{\left({{\textbf{Z}}}/q {{\textbf{Z}}}\right)^\times}}$ . Also, $f \in {{\textbf{Z}}}[X] \setminus \{0\}$ and $\deg f < \varphi(d)$ . Thus, we can use Proposition 1·15 to bound the modulus of the inner sum for any index i such that $|H_q^{(i)} | \geqslant q^\delta$ (and there exists at least one such index i thanks to the growth condition (4·4)).

We deduce that there exists an integer $N_f$ such that for all $q > N_f$ such that q is d-admissible, we have:

\begin{equation*} \frac{1}{|H_q^{(i)}|} \left| \sum_{a_i \in H_q^{(i)}}^{} e\left( \frac{a_if( w_q^{m_i})}{q} \right) \right| \ll_{f, \delta} \frac{1}{q^\varepsilon} \quad \text{ where } \varepsilon = \varepsilon(\delta) > 0\end{equation*}

for at least one $i \in \{1, \dots, n\}$ , which may vary with q. As the other factors in the product (4·6) have absolute value bounded above by one, we may conclude that the whole product has its modulus bounded above by $1/ q^\varepsilon$ (up to multiplicative constants), hence the conclusion.

Conclusion of the proof of Theorem A. In this section, we proved Proposition 2·2 and Proposition 2·5, and each of them implies one of the cases of Theorem A thanks to Propositions 2·4 and 2·6. This completes the proof of our main result.

4·3. Remarks on Theorem A and connexion with previous results

Remark 4·2. There is one last refinement that we did not discuss so far but that can be observed through the proofs: it is the fact that one can fix some of the coefficients $a_i$ when one studies the equidistribution of the restricted sums

\begin{equation*}\sum_{\substack{x \in ({{{{\textbf{Z}}}/q {{\textbf{Z}}}}})^\times \\ x^d = 1}} e\left(\frac{a_1 x^{m_1} + \dots + a_n x^{m_n}}{q}\right)\end{equation*}

in the case where the $m_i$ are coprime with d. This remark is not included in Theorem A (b) because the statement was already quite long without it.

Let $s \in \{1, \dots, n\}$ and let $\{i_1, \dots, i_s\} \subseteq \{1, \dots, n\}$ . We fix $n-s$ integers $a_i$ for $i \in \{1, \dots, n\} \setminus \{i_1, \dots, i_s\}$ . Then the sets of sums:

(4·7) \begin{equation} \left\{ \sum_{\substack{x \in ({{{{\textbf{Z}}}/q {{\textbf{Z}}}}})^\times \\ x^d = 1}} e\left(\frac{a_1 x^{m_1} + \dots + a_n x^{m_n}}{q}\right); \ (a_{i_1}, \dots,a_{i_s}) \in H_q^{(i_1)} \times \dots \times H_q^{(i_s)} \right\} \end{equation}

(in other words, the ones where only some of the $a_i$ are free) become equidistributed in the image of $g_d$ with respect to the same measure as in Theorem A (b) provided there exists $\delta > 0$ such that

(4·8) \begin{equation} \forall q \in {\mathcal A}_d, \exists i \in \{i_1, \dots, i_s \}, \quad |H_{q}^{(i)}| \geqslant q^\delta. \end{equation}

Indeed, when one reduces this question to an equidistribution result modulo 1 as in section 2·3, one needs to prove that the sets

\begin{multline*} \bigg\{ \left(\frac{a_1 ( w_q^{m_1})^0 + \dots + a_n ( w_q^{m_n})^0}{q}, \dots, \frac{a_1 ( w_q^{m_1})^{\varphi(d)-1} + \dots + a_n ( w_q^{m_n})^{\varphi(d)-1}}{q} \right); \\ (a_{i_1}, \dots, a_{i_s}) \in H^{(i_1)}_q \times \dots \times H^{(i_s)}_q \bigg \} \end{multline*}

become equidistributed modulo 1. After applying Weyl’s criterion, one gets a factor with complex absolute value equal to 1, multiplied by the product

(4·9) \begin{equation} \prod_{i \in \{i_1, \dots,i_s\}}^{} \left[ \frac{1}{|H_q^{(i)}|} \sum_{a_i \in H_q^{(i)}}^{} e\left( \frac{a_if( w_q^{m_i})}{q} \right) \right] \end{equation}

which tends to zero under condition (4·8), thanks to Proposition 1·15.

Remark 4·3. If we take the subgroups $H_q^{(i)}$ as large as possible, Theorem A gives an equidistribution result for the sets of sums

\begin{equation*}\left\{ \sum_{\substack{x \in ({{{{\textbf{Z}}}/q {{\textbf{Z}}}}})^\times \\ x^d = 1}} e\left(\frac{a_1 x^{m_1} + \dots + a_n x^{m_n}}{q}\right); \ a_1, \dots, a_n \in {{\left({{\textbf{Z}}}/q {{\textbf{Z}}}\right)^\times}} \right\}.\end{equation*}

However, to compare our result to the ones already proved in [ Reference Burkhardt, Chan, Currier, Garcia, Luca and Suh4 , Reference Duke, Garcia and Lutz7 , Reference Garcia, Hyde and Lutz10 ], we would like to have the $a_i$ varying in ${{{{\textbf{Z}}}/q {{\textbf{Z}}}}}$ instead of ${{\left({{\textbf{Z}}}/q {{\textbf{Z}}}\right)^\times}}$ . This is the content of Proposition B. In order to obtain this proposition, the first step consists in following the strategy of Section 2 to reduce to two statements on equidistribution modulo 1, corresponding to the two cases (a) and (b) of the proposition. These two statements are the exact analogues of Proposition 2·2 and Proposition 2·5, with the condition $(a_1, \dots, a_n) \in H_q^{(1)} \times \dots \times H_q^{(1)}$ replaced by $(a_1, \dots, a_n) \in \left({{{{\textbf{Z}}}/q {{\textbf{Z}}}}}\right)^n$ . They are proved using Weyl’s criterion, but the convergence towards zero is easier to obtain, since Lemma 1·13 replaces the use of Proposition 1·15. For instance, Proposition B (b) reduces to the convergence towards zero of the following product

(4·10) \begin{equation} \prod_{i \in \{i_1, \dots,i_s\}}^{} \left[ \frac{1}{q} \sum_{a_i \in {{{{\textbf{Z}}}/q {{\textbf{Z}}}}}}^{} e\left( \frac{a_if(w_q^{m_i})}{q} \right) \right], \end{equation}

where f is a non-zero polynomial in ${{\textbf{Z}}}[X]$ with $\deg f < \varphi(d)$ . This product is to Proposition B (b) what (4·9) was to Theorem A (b), in its extended form of the previous remark. Now, thanks to Lemma 1·13 each of the factors in (4·10) is eventually equal to zero as $q \in {\mathcal A}_d$ goes to infinity, and this finishes the proof. Proposition B (a) can be proved by a similar adaptation of the proof of Theorem A (a).

This proposition generalises the previous results obtained in [ Reference Burkhardt, Chan, Currier, Garcia, Luca and Suh4 , Reference Duke, Garcia and Lutz7 , Reference Garcia, Hyde and Lutz10 ]. In these articles, only sums of the type

\begin{equation*}\sum_{x^d = 1}^{} e\left( \frac{ax}{q}\right) \quad \text{ or } \quad \sum_{x^d = 1}^{} e\left( \frac{ax+b x^{-1}}{q}\right)\end{equation*}

were considered. In Proposition B (b), we prove that the equidistribution theorems obtained in loc. cit. extend to families of sums with a more general numerator inside the exponentials, namely Laurent polynomials of the form $a_1 x^{m_{1}}+ \dots + a_n x^{m_n}$ , as soon as the exponents $m_i$ are coprime with d. Moreover, it also explains that one can fix some of the coefficients $a_i$ . As long as one of them varies in all ${{{{\textbf{Z}}}/q {{\textbf{Z}}}}}$ , the equidistribution result will hold. In particular, the equidistribution results obtained in [ Reference Burkhardt, Chan, Currier, Garcia, Luca and Suh4 ] for sets of Kloosterman sums

\begin{equation*}\mathcal K_q(-,-,d) \,:\!=\, \left\{ {{\textrm{K}}}_q(a,b, d)\,:\!=\, \sum_{x^d = 1}^{} e\left( \frac{ax+b x^{-1}}{q}\right); \ a,b \in ({{{{\textbf{Z}}}/q {{\textbf{Z}}}}})^2 \right\}\end{equation*}

also hold if one fixes an integer $a_0$ , and considers the sets

\begin{equation*}\mathcal K_q(a_0,-,d) \,:\!=\, \left\{ {{\textrm{K}}}_q(a_0,b, d); \ b \in {{{{\textbf{Z}}}/q {{\textbf{Z}}}}} \right\}.\end{equation*}

Finally, case (a) treats the case where some of the $m_i$ may share prime factors with d, and provides the appropriate Laurent polynomial whose image determines the region of equidistribution.

5·1. Laurent polynomials and hypocycloids

So far, the region of the complex plane in which the sums become equidistributed has only been described as the image of a torus via a Laurent polynomial. However, in some cases, one can give a more explicit description of this image, as it was observed in [ Reference Burkhardt, Chan, Currier, Garcia, Luca and Suh4 , Reference Garcia, Hyde and Lutz10 ]. In this section, we review some cases where the regions of equidistribution can be described in geometric terms, without referring to the Laurent polynomials $g_d$ .

The main ingredient is the following lemma, which states that the region ${\mathbb{H}}_d$ from Definition 1·5 is precisely the image of a torus via a specific Laurent polynomial.

Lemma 5·1. Let $d \geqslant 2$ . The image of the map:

\begin{equation*}\begin{array}{ccccc} f & : & {{\mathbb{T}}}^{d-1} & \longrightarrow & {{\textbf{C}}} \\ && (z_1, \dots, z_{d-1})& \longmapsto & z_1 + \dots+z_{d-1} + \frac{1}{z_1 \dots z_{d-1}} \end{array} \end{equation*}

is the region ${\mathbb{H}}_d$ from Definition 1·5.

Proof. See [ Reference Cooper6 , theorem 3·2·3] or [ Reference Kaiser11 , section 3].

The above Laurent polynomial arises in our equidistribution results in the following way: when d is a prime number, the $d{\text{th}}$ cyclotomic polynomial $\phi_d$ is equal to $X^{d-1} + \dots + X +1$ , and this explicit formula allows one to compute the coefficients $c_{j,k}$ which appear in the definition of $g_d$ (Definition 1·2). This exactly gives the Laurent polynomial of the previous lemma, thus allowing us to have a more concrete description of the region of equidistribution in our results. Precisely, this gives:

Proposition 5·3 ([ Reference Garcia, Hyde and Lutz10 , proposition 1]). Let d be a prime number. The polynomial $g_d$ from Definition 1·2 is given by

\begin{equation*} \begin{array}{ccccc} g_d&:& {{\mathbb{T}}}^{\varphi(d)} = {{\mathbb{T}}}^{d-1}& \longrightarrow & {{\textbf{C}}} \\ && (z_1,\dots, z_{d-1}) & \longmapsto & z_{1}+\cdots +z_{d-1}+\dfrac{1}{z_{1}z_{2}\ldots z_{d-1}} \end{array}\end{equation*}

and the image of ${{\mathbb{T}}}^{d-1}$ via $g_d$ is the region ${\mathbb{H}}_d$ from Definition 1·5. In particular, the region of the complex plane in which the sums restricted to the subgroup of order d become equidistributed in Theorem A (b) and Proposition B (b) is ${\mathbb{H}}_d$ .

This proposition relies mostly on the fact that when d is a prime number, we have an explicit formula for the $d^{\text{th}}$ cyclotomic polynomial. As there is also an explicit formula for the $d^{\text{th}}$ cyclotomic polynomial when $d= r^b$ is a prime power, namely

\begin{equation*}\phi _{r^b}\left( X\right) =\sum ^{r-1}_{j=0}X^{jr^{b-1}} \left( = \phi_r\left(X^{r^{b-1}}\right)\right),\end{equation*}

it is not surprising that our understanding of the image of $g_d$ can also be improved in that case. In fact, the explicit formula above leads to the following proposition.

Proposition 5·4 ([ Reference Garcia, Hyde and Lutz10 , corollary 1]). Let $d\,:\!=\, r^b$ be a power of a prime number r. The polynomial $g_d$ from Definition 1·2 is given by

\begin{equation*} \begin{array}{ccccc} g_d&:& {{\mathbb{T}}}^{\varphi(d)} = {{\mathbb{T}}}^{(r-1)r^{b-1}}& \longrightarrow & {{\textbf{C}}} \\ && (z_1,\dots, z_{(r-1)r^{b-1}}) & \longmapsto & \displaystyle \sum ^{(r-1)r^{b-1} }_{j=1}z_{j}+\sum ^{r^{b- 1}}_{m=1}\prod ^{r-2}_{\ell=0}z_{m+\ell r^{b-1}}^{-1} \end{array}\end{equation*}

and the image of ${{\mathbb{T}}}^{\varphi(d)}$ via $g_d$ is the Minkowski sum

\begin{equation*}\sum_{j=1}^{r^{b-1}} {\mathbb{H}}_r \,:\!=\, \left\{ \xi_1 + \cdots + \xi_{r^{b-1}}; \ \xi_1, \dots , \xi_{r^{b-1}} \in {\mathbb{H}}_r \right\}.\end{equation*}

In particular, the region of the complex plane in which the sums restricted to the subgroup of order d become equidistributed in Theorem A (b) and Proposition B (b) is $\displaystyle \sum_{j=1}^{r^{b-1}} {\mathbb{H}}_r.$

Example 5·5. For instance, as it is done in [ Reference Burkhardt, Chan, Currier, Garcia, Luca and Suh4 , theorem 10], for $r=3$ and $b=2$ we have:

\begin{equation*}g_9(z_1, \dots, z_6) =\underbrace{ z_{1}+z_{4}+\dfrac{1}{z_{1}z_{4}}}_{\in {\mathbb{H}}_3}+ \underbrace{z_{2}+z_{5}+\dfrac{1}{z_{2}z_{5}}}_{\in {\mathbb{H}}_3}+\underbrace{z_{3}+z_{6}+\dfrac{1}{z_{3}z_{6}}}_{\in {\mathbb{H}}_3} \cdot\end{equation*}

A drawing of the region of the complex plane ${\mathbb{H}}_3 + {\mathbb{H}}_3 + {\mathbb{H}}_3$ can be found in [ Reference Garcia, Hyde and Lutz10 , figure 11].

5·2. Illustration of Theorem A

We fix d to be equal to 5, and we consider the following sets of Kloosterman sums restricted to the subgroup of order 5:

(5·1) \begin{equation} \left\{ \sum_{\substack{x \in ({{{{\textbf{Z}}}/q {{\textbf{Z}}}}})^\times \\ x^5 = 1}} e\left(\frac{a x + b x^{-1}}{q}\right); \ (a,b) \in H_q^{(1)} \times H_q^{(2)} \right\}\end{equation}

for increasing 5-admissible values of q, and where $H_q^{(1)}$ and $H_q^{(2)}$ denote subgroups of ${{\left({{\textbf{Z}}}/q {{\textbf{Z}}}\right)^\times}}$ . Then Theorem A (b), combined with the geometric interpretation of Proposition 5·3, states that these sets become equidistributed in the region delimited by a 5-cusp hypocycloid, with respect to some measure (obtained as the pushforward measure, under the Laurent polynomial $g_5$ , of the Haar measure on ${{\mathbb{T}}}^4$ ) provided the subgroups $H_q^{(1)}$ and $H_q^{(2)}$ of ${{\left({{\textbf{Z}}}/q {{\textbf{Z}}}\right)^\times}}$ satisfy the following growth condition:

there exists $\delta > 0$ such that:

\begin{equation*}\forall q \in {\mathcal A}_d, \quad |H_q^{(1)} |\geqslant q^\delta \text{ or } |H_q^{(2)}| \geqslant q^\delta .\end{equation*}

In the Fig. 4, $H_q^{(2)}$ is always chosen to be the trivial multiplicative subgroup, and $|H_q^{(1)}| \geqslant q^{1/2}$ .

Fig. 4. The sets of the form (5·1) for three 5-admissible integers q and for the indicated choice of subgroups $H_q^{(1)}$ , $H_q^{(2)}.$

5·3. Illustrations of Proposition B

First, let us illustrate Proposition B (b) in the case where d is a prime number, with the new insight brought by Proposition 5·3.

Let d be a prime number. For all d-admissible integers q, we consider the sums

\begin{equation*}{{\textrm{S}}}_q(a,d)\,:\!=\,\sum_{\substack{x \in ({{{{\textbf{Z}}}/q {{\textbf{Z}}}}})^\times \\x^d = 1}}^{} e\left(\frac{ax}{q}\right) \text{ for } a \in {{{{\textbf{Z}}}/q {{\textbf{Z}}}}}.\end{equation*}

Since the vector ${{\textbf{m}}} =(1) \in {{\textbf{Z}}}^1$ is coprime with d, Proposition B (b) states that these sums all belong to the image of ${{\mathbb{T}}}^{d-1}$ via $g_d$ , and that the sets:

\begin{equation*}\mathcal S_q(-,d) \,:\!=\, \left\{ {{\textrm{S}}}_q(a,d); \ a \in {{{{\textbf{Z}}}/q {{\textbf{Z}}}}} \right\}\end{equation*}

become equidistributed in this image, with respect to the pushforward measure of the Haar measure on ${{\mathbb{T}}}^{d-1}$ . Now, the new insight given by Proposition 5·3 is the interpretation of the image of $g_d$ as the region ${\mathbb{H}}_d$ delimited by a d-cusp hypocycloid.

We illustrate this statement in the case $d=3$ . In the Fig. 5, the points are the sets $\mathcal S_q(-,3)$ while the delimiting curve is the 3-cusp hypocycloid from Definition 1·4.

However, Proposition B (b) covers many other families of exponential sums. For instance, if one takes ${{\textbf{m}}}$ to be the vector $(1,-1) \in {{\textbf{Z}}}^2$ , the same equidistribution phenomenon happens. Indeed, ${{\textbf{m}}} =(-1,1)$ is coprime with d (for any d) and so the proposition also predicts that the sets of sums

\begin{equation*}\mathcal K_q(-,-,d) \,:\!=\, \left\{ {{\textrm{K}}}_q(a,b,d) \,:\!=\, \sum_{\substack{x \in ({{{{\textbf{Z}}}/q {{\textbf{Z}}}}})^\times \\ x^d = 1}} e\left(\frac{ax + bx^{-1}}{q}\right), \quad a,b \in {{{{\textbf{Z}}}/q {{\textbf{Z}}}}} \right\} \end{equation*}

become equidistributed in ${\mathbb{H}}_d$ as q goes to infinity among the d-admissible integers (and with respect to the same measure as in the previous example). Figure 3 of the introduction illustrates this result in the case $d= 5$ . In the case where $d= 3$ , the comparison of the Figs. 5 and 6 illustrates this striking similitude of behaviour for different types of exponential sums, when restricted to subgroups.

Fig. 5. The sets $\mathcal S_q(-,d)$ for $d= 3$ and three 3-admissible values of q.

Fig. 6. The sets $\mathcal K_q(-,-,d)$ for $d= 3$ and three 3-admissible values of q.

Proposition B (b) also states that one can fix a and let b vary in all ${{{{\textbf{Z}}}/q {{\textbf{Z}}}}}$ , and vice-versa. For instance, if we take the previous example of restricted Kloosterman sums and choose to fix $a=1$ and let only b vary, the sets of sums:

\begin{equation*}\mathcal K_q(1,-,d) \,:\!=\, \{ {{\textrm{K}}}_q(1,b,d); \ b\in {{{{\textbf{Z}}}/q {{\textbf{Z}}}}} \}\end{equation*}

will become equidistributed in the same hypocycloid as before, with respect to the same measure.

Thus, Proposition B allows us to recover the equidistribution results from [ Reference Burkhardt, Chan, Currier, Garcia, Luca and Suh4 , Reference Duke, Garcia and Lutz7 , Reference Garcia, Hyde and Lutz10 ] and extends [ Reference Burkhardt, Chan, Currier, Garcia, Luca and Suh4 , theorem 7 and theorem 10] to all values of d, and to the case where only one of the two parameter a and b varies.

Moreover, Proposition B (b) widely generalises the previously known results to other families of exponential sums. Indeed, sums with ax or $ax+bx^{-1}$ inside the exponentials may now be replaced by sums with $a_1 x^{m_1} + \cdots a_n x^{m_n}$ inside the exponentials, provided the $m_i$ are coprime with d. For instance, one can consider the sums

\begin{equation*}{{\textrm Q}}_q(a,b,c,d) \,:\!=\, \sum_{\substack{x \in ({{{{\textbf{Z}}}/q {{\textbf{Z}}}}})^\times \\ x^d = 1}} e\left(\frac{ax^4 + bx^2 +cx}{q}\right) \text{ for } a,b,c \in {{{{\textbf{Z}}}/q {{\textbf{Z}}}}}, \end{equation*}

for all d-admissible integers q. As soon as d is odd, it is coprime with the exponents of X that appear in the polynomials of the form

\begin{equation*}aX^4 + bX^2 + cX.\end{equation*}

Therefore, Proposition B (b) applies to this family of sums, as long as the summation is restricted to a subgroup of odd order. So if we look at the case $d=3$ and we draw the sets

\begin{equation*}\mathcal Q_q(-,-,-,3) = \{ {{\textrm Q}}_q(a,b,c,3); \ a,b,c \in {{{{\textbf{Z}}}/q {{\textbf{Z}}}}} \} \end{equation*}

for different 3-admissible values of q, we observe the same equidistribution as for the other types of sums, inside a 3-cusp hypocycloid (see Fig. 7).

Fig. 7. The sets $\mathcal Q_q(-,-,-,d)$ for $d= 3$ and three 3-admissible values of q.

One could also want to consider sets of Birch sums restricted to a subgroup, that is:

\begin{equation*}{{\textrm{B}}}_q(a,b,d) \,:\!=\, \sum_{\substack{x \in ({{{{\textbf{Z}}}/q {{\textbf{Z}}}}})^\times\\ x^d =1} }^{} e\left(\frac{ax^3 + bx}{q}\right) \text{ o} \unicode{x00F9} a,b \in {{{{\textbf{Z}}}/q {{\textbf{Z}}}}}.\end{equation*}

For instance if we take $d=7$ and look at the sets $ \mathcal B_q(-,-,7) \,:\!=\,\left\{ {{\textrm{B}}}_q(a,b,7) ; \ a,b \in {{{{\textbf{Z}}}/q {{\textbf{Z}}}}} \right\}$ , then Proposition B (b) (combined with Proposition 5·3) states that they should become equidistributed in ${\mathbb{H}}_7$ (the region delimited by a 7-cusp hypocycloid) as q goes to infinity among the 7-admissible integers. This is indeed what the Fig. 8 suggest:

Fig. 8. The sets $\mathcal B_q(-,-,d)$ for $d= 7$ and three 7-admissible values of q.

On the other hand, one could want to consider Birch sums restricted to the subgroup of order 3, that is sums of the type:

\begin{equation*}{{\textrm{B}}}_q(a,b,3) \,:\!=\, \sum_{\substack{x \in ({{{{\textbf{Z}}}/q {{\textbf{Z}}}}})^\times\\ x^3 =1} }^{} e\left(\frac{ax^3 + bx}{q}\right) \text{ o}\unicode{x00F9} a,\ b \in {{{{\textbf{Z}}}/q {{\textbf{Z}}}}}.\end{equation*}

However, this type of sum does not fall inside the range of application of Proposition B (b), because the exponent 3 in the polynomial expression $ax^3+bx$ is not coprime with the order of the subgroup. In fact, this situation is part of case (a) of Proposition B.

Finally, let us illustrate Proposition 5·4, which gives a geometric interpretation of the image of $g_d$ when d is a prime power.

If we take again the example of Kloosterman sums, Proposition B (b) states that the sums:

\begin{equation*} {{\textrm{K}}}_q(a,b,9) = \sum_{\substack{x \in ({{{{\textbf{Z}}}/q {{\textbf{Z}}}}})^\times \\ x^9 = 1}} e\left(\frac{ax + bx^{-1}}{q}\right), \quad a,b \in {{{{\textbf{Z}}}/q {{\textbf{Z}}}}} \end{equation*}

become equidistributed in the image of $g_9$ , with respect to the pushforward measure of the Haar measure on ${{\mathbb{T}}}^{6}$ . Now, thanks to Proposition 5·4 we can interpret the image of $g_9$ as the Minkowski sum of three copies of ${\mathbb{H}}_3$ (see also example 5·5). Therefore, we should observe equidistribution in a region of the shape given by [ Reference Garcia, Hyde and Lutz10 , figure 11]. Figure 9 illustrates this asymptotic behaviour.

Fig. 9. The sets $\mathcal K_q(-,-,9) \,:\!=\, \left\{ {{\textrm{K}}}_q(a,b,9); \ a,b \in {{{{\textbf{Z}}}/q {{\textbf{Z}}}}} \right\}$ for three 9-admissible values of q.