A numerical note on upper bounds for b 2 [g] sets

TL;DR

This paper improves upper bounds on the size of B 2 [g] sets, a generalization of Sidon sets, using numerical methods for cases g = 2, 3, 4, and 5, advancing understanding of their maximal sizes.

Contribution

It provides improved numerical upper bounds for B 2 [g] sets for specific g values, enhancing previous estimates through computational techniques.

Findings

Enhanced upper bounds for B 2 [g] sets with g=2,3,4,5

Numerical methods effectively refine size estimates of these sets

Results contribute to the understanding of additive combinatorics constraints

Abstract

Sidon sets are those sets such that the sums of two of its elements never coincide. They go back to the 30s when Sidon asked for the maximal size of a subset of consecutive integers with that property. This question is now answered in a satisfactory way. Their natural generalization, called B 2 [g] sets and defined by the fact that there are at most g ways (up to reordering the summands) to represent a given integer as a sum of two elements of the set, are much more difficult to handle and not as well understood. In this article, using a numerical approach, we improve the best upper estimates on the size of a B 2 [g] set in an interval of integers in the cases g = 2, 3, 4 and 5.