Large gaps between Romanoff numbers

Abstract

Questions concerning representations of integers as the sum of a prime and a power of two go back to the correspondence between Euler and Goldbach in 1752 and to de Polignac's work of 1849. Romanoff proved that the set of integers admitting such a representation has positive lower density. Following the complementary direction studied by Kalmynin and Konyagin, we consider long intervals containing no Romanoff numbers. We use truncated products of divisor sums to derive the bound $G_{\mathcal{R}}(X)\gg\log\log X$ for the longest block in $[1,X]$ containing no integer of the form $p+2^n$, with $p$ prime and $n\ge1$. This improves the bound $G_{\mathcal{R}}(X)\gg\log\log X/\log\log\log X$ of Kalmynin and Konyagin. The key step extends a translate lemma to sets of $O(\log X)$ integer shifts of absolute value at most $X$: auxiliary primes dividing differences of shifts are removed at negligible cost. After preliminary sieving, this allows all remaining shifts to be covered simultaneously. The argument is unconditional and also applies to $p+a^n$ for every fixed integer $a\ge2$.