Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Random exponential sums and lattice points in regions

Faruk Temur, Cihan Sahillioğulları

Abstract

In this article we study two fundamental problems on exponential sums via randomization of frequencies with stochastic processes. These are the Hardy-Littlewood majorant problem, and $L^{2n}(\mathbb{T}), \ n\in \mathbb{N}$ norms of exponential sums, which can also be interpreted as solutions of diophantine equations or lattice points on surfaces. We establish connections to the well known problems on lattice points in regions such as the Dirichlet divisor problem.

Random exponential sums and lattice points in regions

Abstract

In this article we study two fundamental problems on exponential sums via randomization of frequencies with stochastic processes. These are the Hardy-Littlewood majorant problem, and norms of exponential sums, which can also be interpreted as solutions of diophantine equations or lattice points on surfaces. We establish connections to the well known problems on lattice points in regions such as the Dirichlet divisor problem.

Paper Structure

This paper contains 10 sections, 19 theorems, 255 equations.

Table of Contents

  1. introduction
  2. Preliminaries
  3. Stationary processes
  4. Poisson processes
  5. Random walks
  6. Perturbations of powers I
  7. Arithmetic progressions of larger step size
  8. Perturbations of Green-Ruzsa Sets
  9. Perturbations of Powers II: $p=2n$
  10. The Arithmetic Component

Key Result

Theorem 1

Let $\{X_j\}_{j\in \mathbb{N}}$ be a stationary process taking only integer values. Let the probability mass function of the random variables in our process be denoted by $\mu:\mathop{\mathrm{\mathbb{Z}}}\nolimits\rightarrow \mathop{\mathrm{\mathbb{R}}}\nolimits.$ Let $A\subset \mathop{\mathrm{\math Analogues of these results for $p=\infty$ also hold.

Theorems & Definitions (38)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Theorem 6
  • Theorem 7
  • Theorem 8
  • Theorem 9
  • Lemma 1
  • ...and 28 more