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Exponential sums with polynomials and their applications to primes in sparse sets

Lingyu Guo, Victor Zhenyu Guo, Mengyao Jing

TL;DR

This paper extends exponential-sum bounds from monomials to polynomial phases $f(m,n)$ and derives a new upper bound for the two-variable sum $\mathcal{S}$ using the exponent-pair method. It then applies these results to the distribution of Piatetski-Shapiro primes, both in single and intersected sequences, and to iterated PS sequences, achieving asymptotic prime counts under explicit gamma-thresholds. A lattice-method framework (polar lattices) is developed to translate counts of primes in sparse PS intersections into tractable exponential-sum estimates, while Kolesnik’s Fourier-analytic approach handles floor-function complications in the iterated case. The main contributions include improved admissible ranges for primes in intersections, a complete asymptotic for primes in twice-iterated PS sequences under concrete conditions, and a detailed technical treatment bounding the resulting bilinear and multilinear sums. Overall, the work integrates advanced harmonic-analytic tools with lattice counting to address primes in highly sparse polynomial-image sets, advancing understanding of primes in non-standard sequences and their iterates.

Abstract

Exponential sums with monomials are highly related to many interesting problems in number theory and well studied by many literatures. In this paper, we consider the exponential sums with polynomials and prove a new upper bound. As an application, we study the Piatetski-Shapiro sequence of the form $(\lfloor n^c \rfloor)$ where $c > 1$ is not an integer. We improve the admissible range of the asymptotic formula for primes in the intersection of Piatetski-Shapiro sequences. We also study the iterated Piatetski-Shapiro sequence and prove an asymptotic formula for the prime counting function.

Exponential sums with polynomials and their applications to primes in sparse sets

TL;DR

This paper extends exponential-sum bounds from monomials to polynomial phases and derives a new upper bound for the two-variable sum using the exponent-pair method. It then applies these results to the distribution of Piatetski-Shapiro primes, both in single and intersected sequences, and to iterated PS sequences, achieving asymptotic prime counts under explicit gamma-thresholds. A lattice-method framework (polar lattices) is developed to translate counts of primes in sparse PS intersections into tractable exponential-sum estimates, while Kolesnik’s Fourier-analytic approach handles floor-function complications in the iterated case. The main contributions include improved admissible ranges for primes in intersections, a complete asymptotic for primes in twice-iterated PS sequences under concrete conditions, and a detailed technical treatment bounding the resulting bilinear and multilinear sums. Overall, the work integrates advanced harmonic-analytic tools with lattice counting to address primes in highly sparse polynomial-image sets, advancing understanding of primes in non-standard sequences and their iterates.

Abstract

Exponential sums with monomials are highly related to many interesting problems in number theory and well studied by many literatures. In this paper, we consider the exponential sums with polynomials and prove a new upper bound. As an application, we study the Piatetski-Shapiro sequence of the form where is not an integer. We improve the admissible range of the asymptotic formula for primes in the intersection of Piatetski-Shapiro sequences. We also study the iterated Piatetski-Shapiro sequence and prove an asymptotic formula for the prime counting function.
Paper Structure (21 sections, 22 theorems, 233 equations, 1 figure)

This paper contains 21 sections, 22 theorems, 233 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Applications of the exponential sums
  3. Notations
  4. Outlines
  5. Intersection of Piatetski-Shapiro sequences
  6. The iterated Piatetski-Shapiro sequence
  7. Preliminary
  8. Bounds of exponential sums
  9. A lattice method
  10. The intersection of Piatetski-Shapiro sequences
  11. Primes in the intersection of Piatetski-Shapiro sequences
  12. Proof of Proposition \ref{['theorem: Type I']}
  13. Proof of Proposition \ref{['theorem: Type II']}
  14. Proof of Theorem \ref{['theorem: 2']}
  15. The iterated Piatetski-Shapiro sequence
  16. ...and 6 more sections

Key Result

Theorem 1.1

Let $M,N,X$ be positive real numbers such that $X=MN$. Let $E_j,\beta_j,\gamma_j\ (j=1,\cdots,d)$ be real numbers. For every exponent pair $(\kappa,\lambda)$ and any sufficiently small real $\varepsilon>0$ we have where $|a_m|\ll X^{\varepsilon}$, $|b_n|\ll X^{\varepsilon}$ and

Figures (1)

  • Figure 1: The admissible range of $c_1$ and $c_2$

Theorems & Definitions (29)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 2.1
  • Lemma 2.2
  • Lemma 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • Lemma 3.4
  • ...and 19 more