On the Waring-Goldbach problem with almost equal summands

Abstract

We use transference principle to show that whenever $s$ is suitably large depending on $k\ge 2$, every sufficiently large natural number $n$ satisfying some congruence conditions can be written in the form $n=p_1^k+\cdots +p_s^k$, where $p_1,\dots ,p_s\in [x-x^{\theta },x+x^{\theta }]$ are primes, $x=(n/s{)}^{1/k}$ and $\theta =0.525+\epsilon$. We also improve known results for $\theta$ when $k\ge 2$ and $s\ge k^2+k+1$. For example when $k\ge 4$ and $s\ge k^2+k+1$ we have $\theta =0.55+\epsilon$. All previously known results on the problem had $\theta >3/4$.

38 pages