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SUMSETS CONTAINING A TERM OF A SEQUENCE

Part of: Sequences and sets

Published online by Cambridge University Press:  18 September 2023

MIN CHEN Open the ORCID record for MIN CHEN [Opens in a new window]  and
MIN TANG Open the ORCID record for MIN TANG [Opens in a new window]
MIN CHEN
Affiliation:
School of Mathematics and Statistics, Anhui Normal University, Wuhu 241002, PR China e-mail: 2395409935@qq.com
MIN TANG*
Affiliation:
School of Mathematics and Statistics, Anhui Normal University, Wuhu 241002, PR China
*
e-mail: tmzzz2000@163.com
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Abstract

Let $S=\{s_{1}, s_{2}, \ldots \}$ be an unbounded sequence of positive integers with $s_{n+1}/s_{n}$ approaching $\alpha $ as $n\rightarrow \infty $ and let $\beta>\max (\alpha , 2)$. We show that for all sufficiently large positive integers l, if $A\subset [0, l]$ with $l\in A$, $\gcd A=1$ and $|A|\geq (2-{k}/{\lambda \beta })l/(\lambda +1)$, where $\lambda =\lceil {k}/{\beta }\rceil $, then $kA\cap S\neq \emptyset $ for $2<\beta \leq 3$ and $k\geq {2\beta }/{(\beta -2)}$ or for $\beta>3$ and $k\geq 3$.

Keywords

sumsetssequencespowers of an integer

MSC classification

Primary: 11B13: Additive bases, including sumsets
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 109 , Issue 3 , June 2024 , pp. 420 - 428
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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Footnotes

This work was supported by the National Natural Science Foundation of China (Grant Nos. 11071033, 12371003) and the Top Talents Project of Anhui Department of Education (Grant No. gxbjZD05).

References

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