Here is another realm where improvements are small and tend towards an unknown but existing limit $S$ with currently known bounds $1.2748 ≤ S ≤ 1.5098$ where the upper bound seems not too far from the actual value of $S$.

It is about bounds for the supremum of autoconvolutions of nonnegative functions supported in a compact interval. The discrete version of those (i.e. restricting to functions that are piecewise constant) yields optimal functions which are closely related to Sidon sets.

A fascinating article is Improved bounds on the supremum of autoconvolutions. As a teaser, have a look at the diagram on page 10.

Here is another realm where improvements are small and tend towards an unknown but existing limit $S$ with currently known bounds $1.2748 ≤ S ≤ 1.5098$ where the upper bound seems not too far from the actual value of $S$.

It is about the smallest possible supremum of the autoconvolution of a nonnegative function supported in a compact interval. The discrete version of those (i.e. restricting to functions that are piecewise constant) yields optimal functions which are closely related to Generalized Sidon sets.

A fascinating article is Improved bounds on the supremum of autoconvolutions. As a teaser, have a look at the diagram on page 10.
If the condition "nonnegative" on the function is removed, surprisingly the new supremum may be even smaller.