Trigonometric Polynomials with Frequencies in the Set of Squares and Divisors in a Short Interval

IF 16.4 1区化学 Q1 CHEMISTRY, MULTIDISCIPLINARY

Accounts of Chemical Research Pub Date : 2023-12-19 DOI:10.1007/s00041-023-10064-w

引用次数: 0

Abstract

Let $\gamma _0=\frac{\sqrt{5}-1}{2}=0.618\ldots $ . We prove that, for any $\varepsilon >0$ and any trigonometric polynomial f with frequencies in the set $\{n^2: N \leqslant n\leqslant N+N^{\gamma _0-\varepsilon }\}$ , the inequality $$\begin{aligned} \Vert f\Vert _4 \ll \varepsilon ^{-1/4}\Vert f\Vert _2 \end{aligned}$$ holds, which makes a progress on a conjecture of Cilleruelo and Córdoba. We also present a connection between this conjecture and the conjecture of Ruzsa which asserts that, for any $\varepsilon >0$ , there is $C(\varepsilon )>0$ such that each positive integer N has at most $C(\varepsilon )$ divisors in the interval $[N^{1/2}, N^{1/2}+N^{1/2-\varepsilon }]$ .

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在短间隔内具有平方和除数集合中频率的三角多项式

Abstract Let $\gamma _0=\frac\{sqrt{5}-1}{2}=0.618\ldots $ .我们证明，对于任意的（varepsilon >0\）和任意的三角多项式f，其频率在集合（{n^2: N +N^{gamma _0-\varepsilon }\} ）中，不等式为$$\begin{aligned}。\Vert f\Vert _4 \ll \varepsilon ^{-1/4}\Vert f\Vert _2 \end{aligned}$$成立，这在 Cilleruelo 和 Córdoba 的猜想上取得了进展。我们还提出了这个猜想与鲁兹萨猜想之间的联系，鲁兹萨猜想断言，对于任意 $\varepsilon >0$, 有 $C(\varepsilon )>；0)，使得每个正整数 N 在区间 \([N^{1/2},N^{1/2}+N^{1/2-\varepsilon }]$中最多有 $C(\varepsilon)$个除数。

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来源期刊

Accounts of Chemical Research 化学-化学综合

CiteScore

31.40

自引率

1.10%

发文量

312

审稿时长

2 months

期刊介绍： Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance. Accounts of Chemical Research replaces the traditional article abstract with an article "Conspectus." These entries synopsize the research affording the reader a closer look at the content and significance of an article. Through this provision of a more detailed description of the article contents, the Conspectus enhances the article's discoverability by search engines and the exposure for the research.