Characterize the possible power-sum differences

Characterize the possible sizes of the differences \(\sum_{a\in A}a^j-\sum_{b\in B}b^j\) at degree \(j=k+1\) in Theorems \ref{main_theorem} and \ref{main_theorem_variant}, and at degrees \(j\in\{1,2,\ldots,k\}\setminus J\) in Corollary \ref{wooley_corollary}.

Background

The main theorems establish nonzero differences of the relevant power sums but do not quantify their possible magnitudes. The authors ask for stronger control of these differences because it would sharpen the range estimates in Corollary \ref{smooth_corollary} and related product-separation results.

References

What can one say about the possible sizes of $\sum_{a \in A} aj - \sum_{b \in B} bj$ for $j=k+1$ in Theorems \ref{main_theorem} and \ref{main_theorem_variant}, as well as the value of $j \in {1,2,\ldots,k}\setminus J$ in Corollary \ref{wooley_corollary}? The stronger the conclusion on this, the tighter the range of differences in Corollary \ref{smooth_corollary} and similar sorts of conclusions.

— The Prouhet--Tarry--Escott problem for subsets with small doubling in integral domains  (2609.05061 - Croot et al., 4 Sep 2026) in Section 1, Results from the literature and some further directions