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Sign changes of the partial sums of a random multiplicative function III: Average

Marco Aymone

TL;DR

The paper investigates the average number of sign changes $V(x)$ of the partial sums $M_f(x)$ of a Rademacher random multiplicative function. It introduces a general framework for orthogonal random variables with unit variance and a linearity property in the $L^q$ norms, tied to Harper’s small-moment phenomenon, to deduce positive-probability sign changes in short dilations of the interval. The main achievement is a lower bound on the average number of sign changes: $\mathbb{E}V(x) \ge \kappa \dfrac{\log x}{(\log\log x)^{1/2+\varepsilon}}$, plus a local sign-change probability result in dilated intervals, with a suite of examples showing broad applicability beyond independence. The results connect to Erdős–Hunt type phenomena and extend them to dependent or structured random variable systems, offering a versatile method for analyzing sign-change behavior in arithmetic random walks and related sum processes.

Abstract

Let $V(x)$ be the number of sign changes of the partial sums up to $x$, say $M_f(x)$, of a Rademacher random multiplicative function $f$. We prove that the averaged value of $V(x)$ is at least $\gg (\log x)(\log\log x)^{-1/2-ε}$. Our new method applies for the counting of sign changes of the partial sums of a system of orthogonal random variables having variance $1$ under additional hypothesis on the moments of these partial sums. In particular, we extend to larger classes of dependencies an old result of Erdős and Hunt on sign changes of partial sums of i.i.d. random variables. In the arithmetic case, the main input in our method is the ``\textit{linearity}'' phase in $1\leq q\leq 1.9$ of the quantity $\log \mathbb{E} |M_f(x)|^q$, provided by the Harper's \textit{better than squareroot cancellation} phenomenon for small moments of $M_f(x)$.

Sign changes of the partial sums of a random multiplicative function III: Average

TL;DR

The paper investigates the average number of sign changes of the partial sums of a Rademacher random multiplicative function. It introduces a general framework for orthogonal random variables with unit variance and a linearity property in the norms, tied to Harper’s small-moment phenomenon, to deduce positive-probability sign changes in short dilations of the interval. The main achievement is a lower bound on the average number of sign changes: , plus a local sign-change probability result in dilated intervals, with a suite of examples showing broad applicability beyond independence. The results connect to Erdős–Hunt type phenomena and extend them to dependent or structured random variable systems, offering a versatile method for analyzing sign-change behavior in arithmetic random walks and related sum processes.

Abstract

Let be the number of sign changes of the partial sums up to , say , of a Rademacher random multiplicative function . We prove that the averaged value of is at least . Our new method applies for the counting of sign changes of the partial sums of a system of orthogonal random variables having variance under additional hypothesis on the moments of these partial sums. In particular, we extend to larger classes of dependencies an old result of Erdős and Hunt on sign changes of partial sums of i.i.d. random variables. In the arithmetic case, the main input in our method is the ``\textit{linearity}'' phase in of the quantity , provided by the Harper's \textit{better than squareroot cancellation} phenomenon for small moments of .
Paper Structure (14 sections, 6 theorems, 54 equations)

This paper contains 14 sections, 6 theorems, 54 equations.

Table of Contents

  1. Introduction
  2. Main result and background
  3. A more general Theorem, and examples
  4. Examples
  5. Preliminaries
  6. Notation
  7. Letters appearing throughout the text
  8. Asymptotic notation
  9. Some estimates
  10. Proof of the main result
  11. Setting the definitions
  12. Proof of Theorem \ref{['thorem local sign changes']}
  13. Proof of Corollary \ref{['corollary average']}
  14. Proof of the General Theorem

Key Result

Corollary 1.1

Let $V(x)$ be the number of sign changes of $M_f(u)$ in the interval $u\in [1,x]$. Then there exists a constant $\kappa>0$, such that for each fixed $0<\epsilon<1/100$, for all $x\geq x_0=x_0(\epsilon)$,

Theorems & Definitions (17)

  • Corollary 1.1
  • Remark 1.1
  • Theorem 1.1
  • Theorem 1.2
  • Example 1.1
  • Example 1.2
  • Example 1.3: $\mathcal{B}_2$ or Sidon sets
  • Example 1.4
  • Lemma 2.1
  • proof
  • ...and 7 more