Short intervals for the Romanoff-type sumset

Abstract

Let $X$ be large and let $\mathcal P$ denote the set of primes. Fix positive real parameters $r_1,\dots,r_s$ and a parameter $λ>0$ determined by a balancing relation, and let $\mathcal A_λ(X)\subset[1,2X]$ be the associated lacunary set generated by sums of powers of $2$ with polynomially growing exponents. Set $\mathcal S_λ:=\mathcal P+\mathcal A_λ(X)$. Fix $\varepsilon>0$, choose $θ$ with $2/15+\varepsilon<θ<1-δ_0$, where $δ_0>0$ is an absolute constant, and set $h=X^θ$. We prove that for all but $O_\varepsilon\left(X\exp\left(-c_\varepsilon(\log X)^{1/4}\right)\right)$ values of $x\in[X,2X]$, the short interval $(x,x+h]$ contains between $c_\varepsilon h$ and $C_\varepsilon h$ integers of the form $p+a$ with $p\in\mathcal P$ and $a\in\mathcal A_λ(X)$.

Theorems & Definitions (18)

• Theorem 1.1: Local density of $\mathcal{S}$ in short intervals
• Theorem 1.2: Typical number of representations
• Lemma 1: Guth--Maynard, almost-all short intervals
• Proposition 1: First moment lower bound
• proof
• Definition 1: Prime-pair singular series
• Lemma 2: Selberg upper-bound sieve for prime pairs in short intervals
• Lemma 3: Chen--Ding--Xu--Zhai, small prime factors in differences
• Lemma 4: Average singular series over differences
• proof