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.