A Numerical Note on Upper Bounds for B₂[g] Sets

ABSTRACT

Sidon sets are those sets such that the sums of two of its elements never coincide. They go back to the 1930s 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 B₂[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, is 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 B₂[g] set in an interval of integers in the cases g = 2, 3, 4, and 5.

Keywords:

• numerical approach
• Sidon sets
• upper bounds

2000 AMS Subject Classification:

• 11B34
• (11B75, 11P70)

Additional information

Funding

Both authors are supported by the ANR grant Cæsar, number ANR 12 - BS01 - 0011.