Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Note on a problem of Sárközy on multiplicative representation functions

Yuchen Ding

Abstract

Motivated by a 2001 problem of Sárközy, we classify all situations of the integers $b,c,e$ and $f$ satisfying \begin{align*} \limsup_{n\rightarrow\infty}|d(\mathcal{A},bn+c)-d(\mathcal{A},en+f)|=\infty \end{align*} for any infinite $\mathcal{A}\subset \mathbb{N}$, where $ d(\mathcal{A},m)=\#\{a\in \mathcal{A}:a|m\}. $

Note on a problem of Sárközy on multiplicative representation functions

Abstract

Motivated by a 2001 problem of Sárközy, we classify all situations of the integers and satisfying \begin{align*} \limsup_{n\rightarrow\infty}|d(\mathcal{A},bn+c)-d(\mathcal{A},en+f)|=\infty \end{align*} for any infinite , where

Paper Structure

This paper contains 1 section, 1 theorem, 31 equations.

Table of Contents

  1. Acknowledgments

Key Result

Theorem 1

Let $b,c,e$ and $f$ be integers. Then we have for any infinite $\mathcal{A}\subset \mathbb{N}$ if and only if one of the following cases holds: I. $b=c=e=0$ and $f\neq 0$; or $b=c=f=0$ and $e\neq 0$; II. $b=0$, $e\neq 0$, and $c=0$; or $b\neq 0$, $e= 0$, and $f=0$; III. $b=0$, $e\neq 0$, $c\neq 0$, and $e|f$; or $b\neq 0$, $e= 0$, $f\neq 0$, an

Theorems & Definitions (2)

  • Theorem 1
  • proof : Proof of Theorem \ref{['thm2']}