Determine the correct admissible exponent at the critical moment

Determine the correct order of magnitude of the critical moment \(\int_0^1 |S_k(\alpha;K)|^{1+1/k}\,d\alpha\) for short intervals and thereby determine whether the exponent \(((k+1)/2)\delta_k\) or its \(\Delta_k\)-analogue should be regarded as \((k,1+1/k)\)-admissible.

Background

The paper defines θk,s\theta_{k,s} using the expected moment asymptotics away from the critical exponent s=1+1/ks=1+1/k. At the critical exponent, however, the known complete-interval bounds differ by a substantial logarithmic factor, and the paper proves only short-interval upper and lower bounds of different logarithmic strengths.

Because the correct order of the critical moment is not known, the authors deliberately avoid assigning the resulting interval-length threshold the formal status of a (k,1+1/k)(k,1+1/k)-admissible exponent. Thus, resolving the critical moment’s order would also settle the associated admissibility question.

References

We refrain from labeling the exponent \big(\frac{k+1}{2}\big)\delta_k as (k,1+\frac{1}{k})-admissible, since we do not know the correct order of magnitude to expect for this moment.

— Subconvexity of Short $k$-Free Exponential Sums  (2608.16679 - Doyle, 17 Aug 2026) in Section 1, immediately following Corollary \ref{crit-corollary}

Can one improve the strength of Theorem \ref{thm:Vinogradov} above, so that we get a result like Corollary~\ref{wooley_corollary} just when $|S| \gg_m N\theta$, for some $\theta \in (0,1)$ that does not depend on $k$? For example, can one replace $|S| \gg_m N{1-2/(k2+k+1)}$ with $|S| \gg_m N{1/2}$, say?

— 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