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Trigonometric polynomials with frequencies in the set of cubes

Mikhail R. Gabdullin, Sergei V. Konyagin

Abstract

We prove that for any $ε>0$ and any trigonometric polynomial $f$ with frequencies in the set $\{n^3: N \leq n\leq N+N^{2/3-ε}\}$, one has $$ \|f\|_4 \ll ε^{-1/4}\|f\|_2 $$ with implied constant being absolute. We also show that the set $\{n^3: N\leq n\leq N+(0.5N)^{1/2}\}$ is a Sidon set.

Trigonometric polynomials with frequencies in the set of cubes

Abstract

We prove that for any and any trigonometric polynomial with frequencies in the set , one has with implied constant being absolute. We also show that the set is a Sidon set.
Paper Structure (3 sections, 4 theorems, 28 equations)

This paper contains 3 sections, 4 theorems, 28 equations.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['th1.1']}
  3. Proof of Theorem \ref{['th1.2']}

Key Result

Theorem 1.1

For any $\varepsilon>0$ and any trigonometric polynomial $f$ with frequencies in the set $\{n^3: N \leqslant n\leqslant N+N^{2/3-\varepsilon}\}$,

Theorems & Definitions (5)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • Theorem 2.2
  • Remark 2.3