On monotone increasing representation functions

2023/03/02 by Kiss, Sándor Z., Sándor, Csaba, Yang, Quan-Hui
#11B34 #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2303.01314

Abstract

Let k≥ 2 be an integer and let A be a set of nonnegative integers. The representation function R_A,k(n) for the set A is the number of representations of a nonnegative integer n as the sum of k terms from A. Let A(n) denote the counting function of A.Bell and Shallit recently gave a counterexample for a conjecture of Dombi and proved that if A(n)=o(n^(k-2)/(k)-ε) for some ε>0, then R_{ℕ∖ A,k}(n) is eventually strictly increasing. In this paper, we improve this result to A(n)=O(n^(k-2)/(k-1)). We also give an example to show that this bound is best possible.

Related

• A Dombi Counterexample with Positive Lower Density
• A note on the lower bound of representation functions
• On the Structure of Sets Which Have Coinciding Representation Functions
• On the maximum values of the additive representation functions
• Partitions of the set of nonnegative integers with the same\n representation functions