Sumsets in the set of squares

IF 0.6 4区 数学 Q3 MATHEMATICS
Christian Elsholtz, Lena Wurzinger
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引用次数: 0

Abstract

We study sumsets $\mathcal{A}+\mathcal{B}$ in the set of squares $\mathcal{S}$ (and, more generally, in the set of kth powers $\mathcal{S}_k$, where $k\geq 2$ is an integer). It is known by a result of Gyarmati that $\mathcal{A}+\mathcal{B}\subset \mathcal{S}_k \cap [1,N]$ implies that $\min(|\mathcal{A}|,|\mathcal{B}|)=O_k(\log N)$. Here, we study how the upper bound on $|\mathcal{B}|$ decreases, when the size of $|\mathcal{A}|$ increases (or vice versa). In particular, if $|\mathcal{A}|\geq C k^{\frac{1}{m}} m (\log N)^{\frac{1}{m}}$, then $|\mathcal{B}|=O_k(m^2 \log N)$, for sufficiently large N, a positive integer m and an explicit constant C > 0. For example, with $m\sim \log \log N$ this gives: If $|\mathcal{A}|\geq C_k \log \log N$, then $|\mathcal{B}|=O_k(\log N (\log \log N)^2)$.
正方形集合中的和集
我们研究平方集合 $\mathcal{S}$ 中的和集 $\mathcal{A}+\mathcal{B}$ (更广义地说,是 kth 幂集合 $\mathcal{S}_k$ 中的和集,其中 $k\geq 2$ 是整数)。由嘉尔马蒂的一个结果可知,$mathcal{A}+\mathcal{B}\subset \mathcal{S}_k \cap [1,N]$ 意味着 $\min(|\mathcal{A}|,|\mathcal{B}|)=O_k(\log N)$ 。在这里,我们将研究当 $||mathcal{A}|$ 的大小增加时,$||mathcal{B}|$ 的上界是如何减小的(反之亦然)。特别是,如果 $|\mathcal{A}|\geq C k^{frac{1}{m}} m (\log N)^{frac{1}{m}}$, 那么 $|\mathcal{B}|=O_k(m^2 \log N)$, 对于足够大的 N、一个正整数 m 和一个显式常数 C > 0。 例如,在 $m\sim \log \log N$ 的情况下,这就给出了:如果$|\mathcal{A}|geq C_k \log \log N$,那么$|\mathcal{B}|=O_k(\log N (\log \log N)^2)$。
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来源期刊
CiteScore
1.30
自引率
0.00%
发文量
36
审稿时长
6-12 weeks
期刊介绍: The Quarterly Journal of Mathematics publishes original contributions to pure mathematics. All major areas of pure mathematics are represented on the editorial board.
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