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

IF 1.2 3区数学 Q2 MATHEMATICS, APPLIED

Journal of Fourier Analysis and Applications 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

Journal of Fourier Analysis and Applications 数学-应用数学

CiteScore

2.10

自引率

16.70%

发文量

审稿时长

6-12 weeks

期刊介绍： The Journal of Fourier Analysis and Applications will publish results in Fourier analysis, as well as applicable mathematics having a significant Fourier analytic component. Appropriate manuscripts at the highest research level will be accepted for publication. Because of the extensive, intricate, and fundamental relationship between Fourier analysis and so many other subjects, selected and readable surveys will also be published. These surveys will include historical articles, research tutorials, and expositions of specific topics. TheJournal of Fourier Analysis and Applications will provide a perspective and means for centralizing and disseminating new information from the vantage point of Fourier analysis. The breadth of Fourier analysis and diversity of its applicability require that each paper should contain a clear and motivated introduction, which is accessible to all of our readers. Areas of applications include the following: antenna theory * crystallography * fast algorithms * Gabor theory and applications * image processing * number theory * optics * partial differential equations * prediction theory * radar applications * sampling theory * spectral estimation * speech processing * stochastic processes * time-frequency analysis * time series * tomography * turbulence * uncertainty principles * wavelet theory and applications