Upper and lower bounds on the size of $B_k[g]$ sets

TL;DR

This paper advances the understanding of the maximum size of $B_k[g]$ sets by improving upper bounds for larger $g$ and $k$, and refining bounds for the case $g=1$, combining theoretical bounds with constructions.

Contribution

It provides new upper bounds for $F_{k,g}(n)$ for large $g$ and $k$, and improves the error term in the case $g=1$, also matching a known lower bound construction.

Findings

Improved upper bounds on $F_{k,g}(n)$ for $g>1$ and large $k$.

Matched the best known upper bound for $g=1$ with an improved error term.

Provided a lower bound matching Lindström's construction, removing a hypothesis.

Abstract

A subset $A$ of the integers is a $B_k[g]$ set if the number of multisets from $A$ that sum to any fixed integer is at most $g$. Let $F_{k,g}(n)$ denote the maximum size of a $B_k[g]$ set in $\{1,\dots, n\}$. In this paper we improve the best-known upper bounds on $F_{k,g}(n)$ for $g>1$ and $k$ large. When $g=1$ we match the best upper bound of Green with an improved error term. Additionally, we give a lower bound on $F_{k,g}(n)$ that matches a construction of Lindstr\"om while removing one of the hypotheses.