On a Problem of Chen and Lev

Abstract

For a given set $S\subset \mathbb{N}$, $R_{S}(n)$ is the number of solutions of the equation $n=s+s^{\prime },s<s^{\prime },s,s^{\prime }\in S$. Suppose that $m$ and $r$ are integers with $m>r\geq 0$ and that $A$ and $B$ are sets with $A\cup B=\mathbb{N}$ and $A\cap B=\{r+mk:k\in \mathbb{N}\}$. We prove that if $R_{A}(n)=R_{B}(n)$ for all positive integers $n$, then there exists an integer $l\geq 1$ such that $r=2^{2l}-1$ and $m=2^{2l+1}-1$. This solves a problem of Chen and Lev [‘Integer sets with identical representation functions’, Integers16 (2016), A36] under the condition $m>r$.

On the integer sets with the same representation functions

“…In 2017, Kiss and Sándor solved Problem 2 affirmatively. Later, Li and Tang [8], Chen, Tang and Yang [3] solved Problem 1 under the assumption 0 ≤ r < m. Recently, Chen and Chen [2] solved Problem 1 affirmatively.…”

Partitions of finite nonnegative integer sets with identical representation functions

“…for all sufficiently large integers n. By using the Thue-Morse sequence, Dombi [6] answered Sárközy's problem affirmatively. In the last few decades, there are many results about representation functions (see, for example, [2,4,5,9,7,10,11,12,13,14,15,16,17]).…”

On Integer Sets With the Same Representation Functions

“…[2] solved Problem 1.1 under the assumption 0 ≤ r < m. Recently, Chen and Chen [1] solved Problem 1.1 affirmatively. In this paper, we consider two sets C and D satisfying C ∪ D = [0, m] \ {r} and C ∩ D = ∅.…”