Uniform bounds for exceptional sums of powers

Determine uniform upper bounds for the number E_{k,s}(N) of positive integers n\leq N that cannot be represented as a sum of exactly s positive k-th powers, for general integers k>1 and s>0.

Background

The paper studies sets A\subset[N] whose difference set avoids sums of s positive k-th powers and proves power-saving upper bounds for |A| when k is odd and s is even, primarily under the restriction s\leq k. An alternative route relates |A| to the exceptional-set quantity E_{k,s}(N), since a set avoiding these differences satisfies |A|\leq E_{k,s}(N)+1.

The authors note that this alternative approach may be especially promising for large s, because sufficiently strong bounds on E_{k,s}(N) would directly yield corresponding bounds for |A|; for sufficiently large s relative to k, it can even imply |A|\ll 1. They explicitly identify obtaining uniform bounds for E_{k,s}(N) as unresolved in general. They give known estimates for k=3 in several even-s cases, including bounded exceptional sets for even s\geq 8.

References

However, the problem of finding uniform bounds on $E_{k, s}(N)$ is open for general $k, s$.

— The Furstenberg-Sárközy theorem for sums of an even number of odd powers  (2609.16595 - Kalogirou et al., 15 Sep 2026) in Introduction, paragraph beginning “An alternative approach”

In the case of cubes, it is conjectured that every sufficiently large integer may be written as the sum of at most four nonnegative cubes.

— Computational results on sums of a prime with squares or cubes  (2609.20505 - Applegate et al., 17 Sep 2026) in Section 1, paragraph 1