On the number of representations of integers as differences between Piatetski-Shapiro numbers
Yuuya Yoshida
TL;DR
This work analyzes representations of integers as differences of Piatetski-Shapiro numbers PS(α) = {⌊n^α⌋}. It derives a sharp asymptotic N_α(d) ∼ β α^{−β} ζ(β) d^{β−1} for 1<α< (√21+4)/5 with β = 1/(α−1), and extends the result to k-term arithmetic progressions within PS(α). It then connects these representations to counts of triples (l,m,n) with ⌊l^α⌋+⌊m^α⌋=⌊n^α⌋ and proves an upper bound E_α(N) ≪_α N^{4−α} for the additive energy in the range 1<α≤4/3, establishing optimality up to logarithmic factors in that range. The methodology combines equidistribution modulo 1, exponential-sum estimates, and discrepancy theory to control error terms and obtain precise constants involving the Riemann zeta function. Overall, the paper advances understanding of the additive structure of PS(α) sequences and provides precise asymptotics for representations by differences and sums of floor-Power terms.
Abstract
For $α>1$, set $β=1/(α-1)$. We show that, for every $1<α<(\sqrt{21}+4)/5\approx1.717$, the number of pairs $(m,n)$ of positive integers with $d=\lfloor{n^α}\rfloor - \lfloor{m^α}\rfloor$ is equal to $βα^{-β}ζ(β)d^{β-1} + o(d^{β-1})$ as $d\to\infty$, where $ζ$ denotes the Riemann zeta function. We use this result to derive an asymptotic formula for the number of triplets $(l,m,n)$ of positive integers such that $l<x$ and $\lfloor{l^α}\rfloor + \lfloor{m^α}\rfloor = \lfloor{n^α}\rfloor$. Furthermore, we prove that the additive energy of the sequence $(\lfloor{n^α}\rfloor)_{n=1}^N$, i.e., the number of quadruples $(n_1,n_2,n_3,n_4)$ of positive integers with $\lfloor{n_1^α}\rfloor+\lfloor{n_2^α}\rfloor=\lfloor{n_3^α}\rfloor+\lfloor{n_4^α}\rfloor$ and $n_1,n_2,n_3,n_4\le N$, is equal to $O_α(N^{4-α})$ when $1<α\le4/3$.