Additive problems on $\lfloor p^c \rfloor$

TL;DR

This paper studies additive problems involving the sequence of floor powers of primes, proving representation results for large integers and analyzing shifted primes within this sequence for certain ranges of the parameter c.

Contribution

It establishes new additive representation results for the sequence of prime powers and investigates the distribution of shifted primes in this sequence for specific ranges of c.

Findings

Any large integer can be expressed as the sum of a prime power and a prime for c in (0, 13/15).

Asymptotic formulas are obtained for shifted primes in the sequence for c in (0, 13/15).

Results hold for almost all fixed positive c outside the integers.

Abstract

The sequence $$ \mathbb{P}^{(c)}=(\lfloor p^c \rfloor)_{p\in \mathbb{P}}\quad (c>0,c\notin \mathbb{N}), $$ is an important subsequence of the well-known Piatetski-Shapiro sequence, where $\mathbb{P}$ is the set of prime numbers and $\lfloor \cdot \rfloor$ is the floor function. We prove that for all $c \in (0, 13/15)$, any large enough integer $N$ can be represented as $$ N=\lfloor p^c\rfloor+q, $$ where $p$ and $q$ are primes. We also prove the result holds for almost all fixed positive $c \in \mathbb{R}\setminus\mathbb{Z}$. Moreover, we investigate shifted primes in this sequence, obtaining an asymptotic formula for all $c \in (0, 13/15)$ and an almost-all result for fixed positive $c \in \mathbb{R}\setminus\mathbb{Z}$.