Waring-Goldbach problem in short intervals

Abstract

Let $k\ge 2$ and $s$ be positive integers. Let $\theta \in (0,1)$ be a real number. In this paper, we establish that if $s>k(k+1)$ and $\theta >0.55$, then every sufficiently large natural number $n$, subjects to certain congruence conditions, can be written as $$n=p_1^k+\cdots +p_s^k,$$ where $p_i(1\le i\le s)$ are primes in the interval $((\frac{n}{s}{)}^{\frac{1}{k}}-n^{\frac{\theta }{k}},(\frac{n}{s}{)}^{\frac{1}{k}}+n^{\frac{\theta }{k}}]$. The second result of this paper is to show that if $s>\frac{k(k+1)}{2}$ and $\theta >0.55$, then almost all integers $n$, subject to certain congruence conditions, have above representation.

accepted version: with the referee's suggestion and a quantitative almost all result (Theorem 3)

References in corpus (3)

• Vinogradov's theorem with almost equal summands
• Four variants of the Fourier-analytic transference principle
• Discorrelation between primes in short intervals and polynomial phases