Existence of asymptotic limits for generalized Sidon-set maxima

Establish whether the normalized extremal quantity \(\beta_g(n)/\sqrt{gn}\) has a limit as \(n\to\infty\) for every fixed positive integer \(g\), rather than merely having separately controlled liminf and limsup.

Background

The function βg(n)\beta_g(n) measures the largest subset of [n][n] whose sum representation function is bounded above by gg. The paper notes that existing work gives asymptotic-order bounds and studies the liminf and limsup of the normalized quantity, but a full limit theorem analogous to the paper’s result for difference bases has been formulated as a conjecture in earlier work. The issue remains an open problem in the sums setting.

References

For example, proving a version of Theorem 1 but for βg (n) was conjectured in [11] and many papers have been written giving bounds on the liminf and lim-sup (Section 1.1 of [11] contains references to much of the progress on β2(n) over the years).

— Cardinalities of $g$-difference sets  (2501.11736 - Schmutz et al., 20 Jan 2025) in Section 5.1, page 15