Finite-coloring property for exponentially growing Hardy functions

Determine whether, for every fixed positive integer k, there exists an exponentially growing Hardy function α such that every k-coloring of the positive integers contains distinct monochromatic positive integers x and y satisfying x+y=⌊α(n)⌋ for some positive integer n.

Background

The paper proves that if the sequence of target values a_n is lacunary, then some finite coloring avoids distinct monochromatic solutions to x+y=a_n. It also constructs, for every fixed number of colors k, lacunary sequences that nevertheless force monochromatic sums, although these sequences are not generally shown to arise from an exponentially growing Hardy function.

The unresolved issue is whether the stronger structural requirement a_n=⌊α(n)⌋, with α an exponentially growing Hardy function, can support the same universal k-coloring phenomenon for every fixed k. This question lies outside the paper’s main sub-polynomial regime and concerns the boundary of its coloring results.

References

Although for k\geq3 this particular construction cannot arise from an exponentially growing Hardy function, we do not know whether, for every fixed k, there exists an exponentially growing Hardy function α such that every k-coloring of \mathbb{N} contains distinct monochromatic x,y satisfying x+y=\lfloor\alpha(n)\rfloor for some n\in\mathbb{N}.

— An ergodic approach to equations of the form $x+y=\lfloorα(n)\rfloor$  (2609.20727 - Bergelson et al., 17 Sep 2026) in Section “Some counterexamples and remarks,” immediately following the discussion of lacunary sequences and finite colorings