2. Structure Theorem for Almost-Kernel Vectors

We begin by showing the easy fact that, except with exponentially small probability, no sparse vector has sparse image under M.

In particular, 0-kernel vectors correspond to the usual right-kernel of M.

Proof. Fix a nonzero $\boldsymbol{v} \in \mathbb{F}_p^n$ with $v_1 \neq 0$ . We begin by computing the probability that $M\boldsymbol{v}$ is $n/4$ -sparse. We denote the rows of M by $R_1,\dots, R_n$ and reveal the rows from bottom-to-top. Since the first entry $R_i$ is independent $R_{i+1},\dots, R_n$ , since $v_1 \neq 0$ , and since $p\geq 3$ , it follows that

\begin{equation*}\max_{R_{i+1},\dots, R_{n}}\mathbb{P}[R_i\cdot \boldsymbol{v} = 0 \mid R_{i+1},\dots, R_{n}] \leq \frac{1}{2}.\end{equation*}

Therefore, the probability that $M\boldsymbol{v}$ is $n/4$ -sparse is at most

\begin{equation*}\frac{1}{2^{n-n/4}}\cdot \binom{n}{n/4}.\end{equation*}

Finally, taking the union bound over the at most

\begin{equation*}\exp(nH(1/(16\log p))p^{n/(16\log p)}\end{equation*}

choices of $n/(16\log{p})$ -sparse vectors in $\mathbb{F}_{p}^{n}$ gives the desired conclusion.

We recall the definition of the atom probability of a vector $\boldsymbol{v} \in \mathbb{F}_{p}^{n}$ with respect to Rademacher random variables.

Definition 2·3. The atom probability of a vector $\boldsymbol{v} \in \mathbb{F}_{p}^{n}$ is defined as

\begin{equation*}\rho_{\mathbb{F}_p}(\boldsymbol{v}) = \max_{r \in \mathbb{F}_p}\mathbb{P}[\xi_1 v_1 + \dots + \xi_n v_n = r],\end{equation*}

where $\xi_1,\dots, \xi_n$ are i.i.d. Rademacher random variables.

We will need the following ‘crude’ structure theorem for $(n-n/\log p)$ -kernel vectors, which follows from a more careful version of the argument in [ Reference Ferber and Jain10 ]. For the reader’s convenience, we include details in Appendix B.

Since

\begin{equation*}\rho_{\mathbb{F}_p}(\boldsymbol{v}) \leq \rho_{\mathbb{F}_p}(\boldsymbol{v}|_S)\end{equation*}

for any $S\subseteq [n]$ , the crude structure theorem shows that any $(n/\log p)$ -kernel vector of $M_n - \lambda I_n$ has atom probability at most $\dfrac{C_{2.4}}{p}$ . While this result is optimal up to the constant $C_{2.4}$ , it is unfortunately insufficient for our application. The next proposition, which is one of the main innovations of this paper, allows us to show that any $(n/\log p)$ -kernel vector of $M_{n}-\lambda I_n$ has many disjoint chunks with atom-probability approximately $1/p$ .

Proposition 2·5. Suppose that $2 < p\le\exp(c_{2.5}n^{1/4})$ and fix $\lambda\in\mathbb{F}_p$ . Let $m = C_{2.5}\log p$ . With probability at least $1-\exp(\!-n/9)$ , every vector $\boldsymbol{v}$ orthogonal to at least $n-n/\log p$ rows of the random symmetric matrix $M_n-\lambda I_n$ over $\mathbb{F}_p$ has the following property, which we denote by ( $\dagger$ ): given any set $T\subseteq[n]$ of size $n/4$ , there at least $n/(2m)$ disjoint sets S of size m in $[n]\setminus T$ such that

\begin{equation*}\operatorname{disc}_{\mathbb{F}_p}(\boldsymbol{v}|_S) = \sup_{x\in\mathbb{F}_p}\bigg|\mathbb{P}_\xi\bigg[\sum_{i\in S}\xi_iv_i = x\bigg]-\frac{1}{p}\bigg|\le\frac{C_{2.5}}{p^2}.\end{equation*}

Proof. This is trivial for $p\le C_{2.5}^{1/2}$ , so assume the opposite. Let $r = n/\log p$ .

First, by Lemma 2·2 and Proposition 2·4, we may assume that $\boldsymbol{v}$ has some $S\subseteq\operatorname{supp}\!(\boldsymbol{v})$ with $|S|\in [\sqrt{n\log n},n^{3/4}]$ such that

\begin{equation*}\rho_{\mathbb{F}_p}(\boldsymbol{v}|_S)\le\frac{C_{2.4}}{p}.\end{equation*}

For some fixed vector with this property, a straightforward argument (cf. proof of Proposition 2·4) shows that the probability that it is orthogonal to some set of $n-r$ rows of $M_n$ is at most

(2·1) \begin{equation}\binom{n}{r}\cdot\bigg(\frac{C_{2\cdot4}}{p}\bigg)^{n-n^{3/4}-r}\le C^np^{-n}.\end{equation}

We will use a union bound argument to establish Proposition 2·5. Given (2·1), the key is to show that the set of vectors $\boldsymbol{v}$ which fail to satisfy ( $\dagger$ ) is sufficiently sparse in $\mathbb{F}_p^n$ . To this end, consider some T of size $n/4$ and for each such T, consider a fixed (but otherwise arbitrary) partition of $[n]\setminus T$ into $S_1,\ldots,S_{3n/(4m)}$ of size m (up to rounding). There are at most $2^n$ ways to choose T and at most $2^n$ ways to choose which of these sets satisfy

\begin{equation*}\operatorname{disc}_{\mathbb{F}_p}\!(\boldsymbol{v}|_{S_i})\le C_{2.5}/p^2.\end{equation*}

If $\boldsymbol{v}$ violates ( $\dagger$ ), then at least a third of these indices are failures. Thus, we see that the number of vectors which violate ( $\dagger$ ) is at most

(2·2) \begin{equation}2^n\cdot2^n\cdot(p^{3n/4})\cdot|U|^{n/(4m)},\end{equation}

where U is the set of $\mathbb{F}_p$ -vectors of size m with

\begin{equation*}\operatorname{disc}_{\mathbb{F}_p}\!(\boldsymbol{v}|_{S_i}) > C_{2.5}/p^2.\end{equation*}

Fix some sufficiently large absolute constant $C' > 0$ (depending on C in (2·1)). We claim that if $C_{2.5}\ge 1$ chosen large enough, then $|U|\le(p/C')^m$ $\Big(\textrm{recall that}\ p > C_{2.5}^{1/2}\Big)$ . Note that this claim, together with (2·1) and (2·2) completes the proof of Proposition 2·5.

We now prove the claim. Note that if $\boldsymbol{b} = (b_1,\ldots,b_m)\in U\subseteq\mathbb{F}_p^m$ for i.i.d. Rademacher random variables $\xi_1,\dots, \xi_n$ we have that

\begin{align*}\frac{1}{p^2} < \operatorname{disc}_{\mathbb{F}_p}\!(\boldsymbol{b}) &= \sup_{x\in\mathbb{F}_p}\bigg|\mathbb{P}_\xi[b_1\xi_1+\cdots+b_m\xi_m=x]-\frac{1}{p}\bigg|\\&= \sup_{x\in\mathbb{F}_p}\bigg|\frac{1}{p}\sum_{\ell\in\mathbb{F}_p^\times}\exp\!(2\pi i(\ell x/p))\prod_{j=1}^m\cos\!(2\pi(\ell b_j/p))\bigg|\\&\le\frac{1}{p}\sum_{\ell\in\mathbb{F}_p^\times}\prod_{j=1}^m|\cos\!(2\pi(\ell b_j/p))|\\&\le\frac{1}{p}\sum_{\ell\in\mathbb{F}_p^\times}\exp\!\bigg(\!-\sum_{j=1}^m\lVert{2\ell b_j/p}\rVert_{\mathbb{R}/\mathbb{Z}}^2\bigg)\\&\le\max_{\ell\in\mathbb{F}_p^\times}\exp\!\bigg(\!-\sum_{j=1}^m\lVert{\ell b_j/p}\rVert_{\mathbb{R}/\mathbb{Z}}^2\bigg).\end{align*}

In particular, there is some $\ell\in\mathbb{F}_p^\times$ with

\begin{equation*}\sum_{j=1}^m\lVert{\ell b_j/p}\rVert_{\mathbb{R}/\mathbb{Z}}^2\le 2\log p.\end{equation*}

To count the number of such vectors, note that there are at most p choices of $\ell$ . Moreover, since coordinate-wise multiplication by $\ell$ is a bijection from $\mathbb{F}_{p}^{m}$ onto itself, it follows that

\begin{equation*}|U| \leq p\cdot\#\{\boldsymbol{b}\in \mathbb{F}_p^m \;:\; \sum_{j=1}^{m}\|b_j/p\|^{2}_{\mathbb{R}/\mathbb{Z}} \leq 2\log{p}\}.\end{equation*}

The second term in the product is a count of lattice points in a ball. A standard volumetric argument shows that there are at most $\operatorname{vol}(B_2^m(R+\sqrt{m}))$ integer lattice points in an m-dimensional ball of radius R. Hence, since $\operatorname{vol}(B_2^m(1))\le A^mm^{-m/2}$ for some absolute constant A we see that

\begin{equation*}|U|\le p\cdot A^m\bigg(1+\frac{p\sqrt{2\log p}}{\sqrt{m}}\bigg)^m\le (2A)^m\cdot\Big(1+p\sqrt{2C_{2.5}^{-1}}\Big)^m\le\Big(8AC_{2.5}^{-1/2}p\Big)^m\end{equation*}

since $m = C_{2.5}\log{p}$ and $p > C_{2.5}^{1/2}$ . Choosing $C_{2.5}$ sufficiently large completes the proof.

3. Proof of Theorem 1·2

We are now in position to deduce Theorem 1·2; the high-level structure of the argument is as in [ Reference Koenig and Nguyen15, Reference Maples18 ]. However, we make explicit a number of details which are implicit in [ Reference Maples18 ] as well as make a number of simplifications and changes to account for use of our structure theorem Proposition 2·5.

We first recall a basic linear algebra fact about full-rank principal minors.

Fix $\lambda\in\mathbb{F}_p$ . We consider the nested sequence of symmetric matrices

\begin{equation*}M_1-\lambda I_1\subseteq M_2-\lambda I_2\subseteq\cdots\subseteq M_n-\lambda I_n,\end{equation*}

where $M_{i} - \lambda I_{i}$ is the top-left $i\times i$ -submatrix of $M - \lambda I$ (hence, the nested sequence is obtained by iteratively adding symmetric “reverse-L” shapes of Rademacher random variables, with a shift by $\lambda$ on the diagonal element). For simplicity, let $A_t = M_t - \lambda I_t$ . Define the events

\begin{equation*}\mathcal{E}_{S,t} = \{A_t \text{ has no nonzero } t/(16\log p)\text{-sparse } t/4\text{-kernel vector}\}, \quad \text{and}\end{equation*}

\begin{equation*}\mathcal{E}_{U,t} = \{\text{every nonzero }(t/\log p)\text{-kernel vector }\boldsymbol{v} \text{ of }A_t \text{ satisfies property (}\dagger\text{)}\},\end{equation*}

where recall that ( $\dagger$ ) is the property that given any set $T\subseteq[t]$ of size $t/4$ , there at least $t/(2m)$ disjoint sets S of size $m = C_{2.5}\log{p}$ in $[t]\setminus T$ such that

\begin{equation*}\operatorname{disc}_{\mathbb{F}_p}\!(\boldsymbol{v}|_S) = \sup_{x\in\mathbb{F}_p}\bigg|\mathbb{P}_\xi\bigg[\sum_{i\in S}\xi_iv_i = x\bigg]-\frac{1}{p}\bigg|\le\frac{C_{2.5}}{p^2}.\end{equation*}

By Lemma 2·2 and Proposition 2·5, we see that

(3·1) \begin{equation}\mathbb{P}[\mathcal{E}_{S,t}^c\vee\mathcal{E}_{U,t}^c]\le 2\exp(\!-t/9).\end{equation}

Lemma 3·2. We have for $2 < p\le\exp(c_{3.2}n^{1/4})$ , fixed $\lambda\in\mathbb{F}_p$ , and for any $k\ge 0$ that

\begin{align*}\mathbb{P}[\!\operatorname{corank}A_{t+1} = k-1|A_{t} \;:\; \operatorname{corank}A_t=k\wedge\mathcal{E}_{S,t}\wedge\mathcal{E}_{U,t}] &= 1 - p^{-k} + O(\!\exp(\!-\Omega(t))),\\\!\!\!\!\!\!\!\mathbb{P}[\!\operatorname{corank}A_{t+1} = k + 0|A_{t} \;:\; \operatorname{corank}A_t=k\wedge\mathcal{E}_{S,t}\wedge\mathcal{E}_{U,t}] & \\ &\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!= p^{-k}-p^{-k-1}+ O(\!\exp(\!-\Omega(t/\log p))),\\\mathbb{P}[\!\operatorname{corank}A_{t+1} = k+1|A_{t}\; :\; \operatorname{corank}A_t=k\wedge\mathcal{E}_{S,t}\wedge\mathcal{E}_{U,t}] &= p^{-k-1} + O(\!\exp(\!-\Omega(t/\log p))).\end{align*}

Before proving Lemma 3·2, let us show how it implies Theorem 1·2.

Proof of Theorem 1·2. We consider the exposure process obtained by iteratively revealing $M_t$ for $1\le t\le n$ and considering the resulting corank of $A_t$ . Starting from a random $A_{n/20}$ , let $\tau$ denote the (random) first value of $t\geq n/20$ such that either (i) $A_{t} \in \mathcal{E}_{S,t}^{c} \cup \mathcal{E}_{U,t}^{c}$ , or (ii) $\operatorname{corank}(A_t) = 0$ . We claim that, except with probability at most $O(\!\exp(\!-\Omega(n/\log{p})))$ , we have $\tau \leq n/2$ , and moreover, $A_{\tau}$ satisfies condition (ii) and not condition (i).

To see this, note that by Lemma 2·2 and Proposition 2·5, the probability that $A_{t} \in \mathcal{E}_{S,t}^{c} \cup \mathcal{E}_{U,t}^{c}$ is $O(\!\exp(\!-\Omega(n)))$ for any $t\geq n/20$ . Moreover, for $A_{t} \in \mathcal{E}_{S,t} \wedge \mathcal{E}_{U,t}$ with $\operatorname{corank}(A_{t}) = k \geq 1$ , we see that $\mathbb{P}[\operatorname{corank}A_{t+1} = k-1 \mid A_{t}] \geq 1-{1}/{3} + O(\!\exp(\!-\Omega(n))) \geq {3}/{5}$ for all n sufficiently large, and similarly, $\mathbb{P}[\!\operatorname{corank}A_{t+1} = k+1 \mid A_{t}] \leq {1}/{8}$ for all n sufficiently large. Therefore, by a straightforward comparison argument, it follows that for all n sufficiently large, the probability that $\tau \leq n/2$ and $A_{\tau}$ satisfies condition (ii) and not condition (i) is at most

\begin{equation*}O(\!\exp(\!-\Omega(n/\log{p}))) + q,\end{equation*}

where $q = O(\!\exp(\!-\Omega(n)))$ is the probability that a biased random walk with steps $-1$ with probability $1/2$ , 0 with probability $1/4$ , and $+1$ with probability $1/4$ and initial state $n/20$ does not hit 0 in $n/2 - n/20$ steps.

To summarise, we have shown that except with probability $O(\!\exp(\!-\Omega(n/\log{p})))$ , there exists some $\tau \in [n/20, n/2]$ such that $A_{\tau} \in \mathcal{E}_{S,\tau} \wedge \mathcal{E}_{U,\tau}$ and $\operatorname{corank}(A_{\tau}) = 0$ . From this point onwards, outside an event of probability at most $O(\!\exp(\!-\Omega(n/\log{p})))$ , it follows by the remark following Lemma 3·2 that we can couple the corank process $A_{\tau+1},\dots, A_{n}$ with the corank process $A^{\prime}_{1},\dots, A^{\prime}_{n-\tau}$ where $A^{\prime}_{i}$ is the top-left $i\times i$ sub-matrix of $M^{\prime}_{n-\tau} - \lambda$ and $M^{\prime}_{n-\tau}$ is an $(n-\tau) \times (n-\tau)$ random symmetric matrix whose independent entries are chosen uniformly from $\mathbb{F}_p$ . By [ Reference Fulman and Goldstein12 , theorem 4·1], for $\tau \in [n/20, n/2]$ and for any $0 \leq k \leq n-\tau$ ,

\begin{equation*}\mathbb{P}[\!\operatorname{corank}A^{\prime}_{n-\tau} = k] = \frac{\prod_{i=0}^{\infty}(1-p^{-(2i+1)})}{\prod_{i=1}^{k}(p^i-1)} + O(p^{-\Omega(n)}),\end{equation*}

which completes the proof.

Finally, we prove Lemma 3·2:

Proof of Lemma 3·2. Note that $\operatorname{rank} A_{t+1}-\operatorname{rank} A_t\in\{0,1,2\}$ since rank is monotone and sub-additive and the rank of the “reverse-L” is at most 2. Since the ambient dimension increases by 1 at each step, it follows that the corank increases by one of the three values $\{0,\pm1\}$ , so that it suffices to prove the first and third equalities. Write

\begin{equation*}A_{t+1} = \begin{bmatrix}A_t&\xi_t\\\xi_t^T&z\end{bmatrix},\end{equation*}

where $\xi_t$ is an i.i.d. Rademacher vector and $z+\lambda$ is an independent Rademacher. Let $\xi$ be the vector $[\xi_t^T z]^T$ .

Step 1. Corank decrease via linear forms. The first equality is only nontrivial when $k\ge 1$ , and in this case we need the rank to increase by 2. Basic linear algebra (cf. [ Reference Costello, Tao and Vu7 , lemma 2·4]) shows that this is equivalent to requiring $\xi_t$ to lie outside the span of the column space of $A_t$ . Equivalently, $\xi_t$ should not be orthogonal to all kernel vectors of $A_t$ (which form a dimension k subspace of $\mathbb{F}_p^t$ ). Let $\boldsymbol{v}_1,\ldots,\boldsymbol{v}_k$ form a basis of the kernel. We have

\begin{align*}\left|\mathbb{P}_\xi[\boldsymbol{v}_j\cdot\xi_t = 0\text{ for all }j\in[k]]-p^{-k}\right| &= \bigg|\frac{1}{p^k}\sum_{\boldsymbol{a}\in\mathbb{F}_p^k\setminus\{\mathbf{0}\}}\mathbb{E}_\xi\exp\bigg(\frac{2\pi i}{p}(\xi_t\cdot(a_1\boldsymbol{v}_1+\cdots+a_k\boldsymbol{v}_k))\bigg)\bigg|\\[6pt] &\le\max_{\boldsymbol{a}\in\mathbb{F}_p^k\setminus\{\boldsymbol{0}\}}\bigg|\mathbb{E}_\xi\exp\bigg(\frac{2\pi i}{p}(\xi_t\cdot(a_1\boldsymbol{v}_1+\cdots+a_k\boldsymbol{v}_k))\bigg)\bigg|.\end{align*}

Let $\boldsymbol{a} = (a_1,\dots, a_k) \in \mathbb{F}_p^{k} \setminus \{\boldsymbol{0}\}$ denote the element attaining the maximum. Since $\boldsymbol{v}_1,\dots, \boldsymbol{v}_k$ are linearly independent vectors in the kernel of $A_t$ , $\boldsymbol{v} = a_1\boldsymbol{v}_1+\cdots+a_k\boldsymbol{v}_k$ is a nonzero kernel vector of $A_t$ . Since we have conditioned on $\mathcal{E}_{S,t}\wedge \mathcal{E}_{U, t}$ , $\boldsymbol{v}$ has at least $t/(16\log p)$ nonzero entries and satisfies property ( $\dagger$ ).

From $\mathcal{E}_{U,t}$ we see that $\boldsymbol{v}$ can be partitioned into at least $t/(2m)$ disjoint sets $S_1,\dots, S_{t/(2m)}$ of size m with $\operatorname{disc}_{\mathbb{F}_p}\!(\boldsymbol{v}|_{S_i})\le C_{2.5}p^{-2}$ for all $i \in [t/(2m)]$ . Hence,

\begin{align*}\mathbb{E}_{\xi}\exp\left(\frac{2\pi i}{p}\xi_t\cdot \boldsymbol{v}\right) &\leq \prod_{\ell=1}^{t/(2m)}\left|\mathbb{E}_{\xi|_{S_\ell}}\exp\left(\frac{2\pi i}{p} \xi|_{S_\ell}\cdot \boldsymbol{v}|_{S_\ell}\right)\right|\\ &= \prod_{\ell=1}^{t/(2m)}\left|\sum_{j \in \mathbb{F}_p}\exp\left(\frac{2\pi i}{p}j\right)\cdot \left(\frac{1}{p} + \bigg(\mathbb{P}_{\xi}\bigg[\sum_{i\in S_{\ell}}v_i\xi_i \equiv j\bmod p\bigg]-\frac{1}{p}\bigg) \right)\right|\\ &\leq \prod_{\ell=1}^{t/(2m)}\left|\sum_{j \in \mathbb{F}_p}\operatorname{disc}_{\mathbb{F}_p}\!(\boldsymbol{v}|_{S_\ell})\right|\\ &\leq (C_{2.5}p^{-1})^{t/(2m)} = \exp(\!-\Omega(t)),\end{align*}

if $p > C_{2.5}$ .

From $\mathcal{E}_{S,t}$ we see that $\boldsymbol{v}$ has support size at least $t/(16\log p)$ , so that

\begin{align*}\mathbb{E}_{\xi}\exp\left(\frac{2\pi i}{p}\xi_t\cdot \boldsymbol{v}\right) &\leq \prod_{j \in \operatorname{supp}\!(\boldsymbol{v})}\left|\mathbb{E}_{\xi_j}\exp\left(\frac{2\pi i}{p}\xi_j v_j\right)\right|\\ &\leq (1-\Omega(1/p^2))^{t/(16\log{p})}\\ &= \exp(\!-\Omega(t/(p^2\log p))) = \exp(\!-\Omega(t)),\end{align*}

if $p\le C_{2.5}$ . Therefore regardless of what $p > 2$ is, we have

\begin{equation*}\mathbb{P}_\xi[\boldsymbol{v}_j\cdot\xi_t = 0\text{ for all }j\in[k]] = p^{-k} + O(\!\exp(\!-\Omega(t))),\end{equation*}

as desired to establish the first equality.

Step 2. Corank increase via quadratic forms. Now we turn to the third equality. First note that it suffices to prove the claim when $k\le t/(64\log p)$ since if $k > t/(64\log p)$ , the first equality already implies that the second and third probabilities are of size $p^{-k} + O(\!\exp(\!-\Omega(t))) = O(\!\exp(\!-\Omega(t)))$ .

By Lemma 3·1, $A_t$ has a principal minor of rank $(t-k)$ . Therefore, without loss of generality, we may suppose that the top left $(t-k)\times(t-k)$ block, call it B, has full rank. Let $\phi$ be the restriction of $\xi$ to these coordinates. In order for the corank of $A_{t+1}$ to be larger than the corank of $A_{t}$ , it must be the case that $\operatorname{rank}(A_{t+1}) = \operatorname{rank}(A_{t})$ . This precisely corresponds to the event

\begin{equation*}\{\exists \boldsymbol{w} \in \mathbb{F}_p^{t-k}\;:\; \xi_{t} = A_{t}\boldsymbol{w}\} \wedge \{\phi^TB^{-1}\phi = z\}.\end{equation*}

Indeed, if $\xi_t$ is not in the column space then the rank of $A_{t+1}$ must increase and if $\phi^TB^{-1}\phi\neq -z$ , then B along with the new elements in the “reverse-L” will have rank $(t-k+1)$ . On the other hand, if the above event holds, then it is easy to see that $\operatorname{rank}(A_{t+1}) = \operatorname{rank}(A_t)$ .

As in the first case, let $\boldsymbol{v}_1,\dots, \boldsymbol{v}_k$ form a basis of the kernel of $A_{t}$ . Then,

\begin{align*}\bigg|\mathbb{P}_\xi[\boldsymbol{v}_j\cdot\xi_t &= 0\text{ for all }j\in[k]\wedge\phi^TB^{-1}\phi = z] - p^{-k-1}\bigg|\\&\le\sup_{\boldsymbol{a}\in\mathbb{F}_p^{k+1}\setminus\{\mathbf{0}\}}\bigg|\mathbb{E}_\xi\exp\bigg(\frac{2\pi i}{p}(\xi_t\cdot(a_1\boldsymbol{v}_1+\cdots+a_k\boldsymbol{v}_k) + a_{k+1}(\phi^TB^{-1}\phi-z))\bigg)\bigg|.\end{align*}

Note here that $\xi_t,\phi,z$ all depend on $\xi$ . Let $\boldsymbol{a} = (a_1,\dots, a_{k+1}) \in \mathbb{F}_p^{k+1}\setminus \{\boldsymbol{0}\}$ denote the element attaining the maximum. If $a_{k+1} = 0$ , we have a bound of $\exp(\!-\Omega(t))$ exactly as in the first case. It therefore suffices to consider $a_{k+1}\neq 0$ . Let $\boldsymbol{v} = a_1\boldsymbol{v}_1+\cdots+a_k\boldsymbol{v}_k$ .

To handle the quadratic term $\phi^T B^{-1} \phi$ , we will use a decoupling trick, which in this context essentially goes back to [ Reference Costello, Tao and Vu7 ]. Let $I\sqcup J = \{1,\ldots,t\}$ be the partition with $J = [t/(64\log p)]$ , and let $\xi_I,\xi_J$ be the obvious restrictions. Let $\xi_J^{\prime}$ be an independent resample of $\xi_J$ . Let $\xi = \xi_I+\xi_J$ and $\xi' = \xi_I+\xi_J^{\prime}$ . Let $\phi' = \xi'\mid_{[t-k]}$ . We have that

\begin{align*}\bigg|\mathbb{E}_\xi&\exp\bigg(\frac{2\pi i}{p}(\xi_t\cdot \boldsymbol{v} + a_{k+1}(\phi^TB^{-1}\phi - z))\bigg)\bigg|^2\\&= \mathbb{E}_{\xi_I,\xi_J,\xi_J^{\prime}}\exp\bigg(\frac{2\pi i}{p}((\xi_J-\xi_J^{\prime})\cdot\boldsymbol{v} + a_{k+1}(\phi^TB^{-1}\phi - (\phi')^TB^{-1}\phi'))\bigg)\\&\leq \mathbb{E}_{\xi_J,\xi_J^{\prime}}\left|\mathbb{E}_{\xi_I}\left[\exp\left(\frac{2\pi i}{p} a_{k+1}(\phi - \phi')^T B^{-1}(\phi + \phi') \right)\bigg | \xi_J, \xi^{\prime}_J\right]\right| \\&\le\mathbb{E}_{\xi_J,\xi_J^{\prime}}\bigg|\mathbb{E}_{\xi_I}\bigg[\exp\bigg(\frac{2\pi i}{p}(2a_{k+1}(\xi_J-\xi_J^{\prime})^TB^{-1}\phi'')\bigg)\bigg|\xi_J,\xi_J^{\prime}\bigg]\bigg|,\end{align*}

where we have abused notation by using $\xi_J - \xi^{\prime}_J$ to denote the extension of this vector to $[t-k]$ with the coordinates in $[t-k]\setminus J$ equal to 0 and where $\phi''$ denotes the $(t-k)$ -dimensional vector which coincides with $\xi$ (and hence $\xi'$ ) on $I \cap [t-k]$ and has remaining coordinates 0.

We now consider two cases. Note that $\mathbb{P}[\xi_J = \xi_J^{\prime}] = 2^{-|J|}$ , which is of size exp $(\!-\Omega(t/\log p))$ . Otherwise, $\boldsymbol{w} = B^{-1}(\xi_J-\xi^{\prime}_{J})$ is a linear combination of at most $t/(64\log p)$ different columns of $B^{-1}$ and hence is orthogonal to at least $(t-k) -t/(64\log p) \geq t/(32\log{p})$ different rows of B. Hence, if we extend $\boldsymbol{w}$ to a t-dimensional vector by padding it with 0s, then the resulting vector is a nonzero vector which is orthogonal to at least $t-t/(32\log{p})$ rows of $A_t$ . Since $A_{t}$ is assumed to satisfy $\mathcal{E}_{U,t}$ , it follows that this vector has at least $t/(2m)$ disjoint sets $S_1,\dots, S_{t/(2m)}$ of size $m = C_{2.5}\log p$ such that $\operatorname{disc}_{\mathbb{F}_p}\!(\boldsymbol{v}|_{S_i})\le C_{2.5}p^{-2}$ for all $i \in [2m]$ . Furthermore $\mathcal{E}_{U,t}$ guarantees that we can take these sets disjoint from the set $J\cup\{t-k+1,\ldots,t\}$ which has size at most $t/(32\log{p})$ . In other words, the sets $S_1,\dots, S_{t/(2m)}$ are contained in $I \cap [t-k]$ , which is the support of the random entries of $\phi''$ . Thus, similar to Step 1, we deduce for realizations of $\xi_J,\xi_J^{\prime}$ such that $\xi_{J} - \xi^{\prime}_{J} \neq 0$ ,

\begin{equation*}\bigg|\mathbb{E}_{\xi_I}\bigg[\exp\bigg(\frac{2\pi i}{p}(2a_{k+1}(\xi_J-\xi_J^{\prime})^TB^{-1}\phi'')\bigg)\bigg|\xi_J,\xi_J^{\prime}\bigg]\bigg|\le(C_{2.5}p^{-1})^{t/(2m)} = \exp(\!-\Omega(t))\end{equation*}

for $p > C_{2.5}$ . For $p\le C_{2.5}$ , we use $\mathcal{E}_{S,t}$ to deduce that $\boldsymbol{w}$ has support size at least $t/(16\log p)$ . Therefore, $\boldsymbol{w}$ has at least $t/(32\log p)$ on the support of the random entries of $\phi''$ , so that the result again follows as in Step 1.