On integers n relatively prime to ⌊ αn2⌋
Abstract
For α∈ℝ and k∈ℕ, let Sα,k(x):=#{n∈ℕ:n≤ x, gcd(n,⌊αnk⌋)=1}. Roth conjectured, and Watson proved, that Sα,1(x)∼ 6x/π2 for every irrational α. Whether the same asymptotic holds for every fixed k≥2 is a special case of a question of Bergelson and Richter, and was asked explicitly by Champagne. We answer this for k=2, with no Diophantine condition on α: for every irrational α, Sα,2(x)∼ x/ζ(2) as x→∞. More precisely, for every fixed A>0 we obtain an error term OA(x/(log q(x))A), where q(x) is the largest continued-fraction convergent denominator of α not exceeding x. The proof sieves Sα,2(x) over divisors, splitting at the parameter R=q(x)1/8. Divisors larger than R are handled by reducing the condition d|gcd(n,⌊αn2⌋) to Möbius sums over arithmetic progressions and applying Zhan's uniform estimate for exponential sums twisted by μ(n) over short intervals; divisors at most R are handled instead through the Erdős-Turán inequality together with Weyl-type bounds for quadratic exponential sums of the form ∑ e(βn2).
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BibTeX
@misc{chatterjee2026integers,
title = {{On integers $n$ relatively prime to $\lfloor αn^2\rfloor$}},
author = {Aahan Chatterjee},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.04345},
eprint = {2611.04345},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.04345},
}
RIS
TY - UNPB
TI - On integers n relatively prime to ⌊ αn^2⌋
AU - Aahan Chatterjee
PY - 2026
DA - 2026/11/03
PB - arXiv
DO - 10.48550/arxiv.2611.04345
UR - https://arxiv.org/abs/2611.04345
AB - For α∈ℝ and k∈ℕ, let S_α,k(x):=#{n∈ℕ:n≤ x, gcd(n,⌊αn^k⌋)=1}. Roth conjectured, and Watson proved, that S_α,1(x)∼ 6x/π^2 for every irrational α. Whether the same asymptotic holds for every fixed k≥2 is a special case of a question of Bergelson and Richter, and was asked explicitly by Champagne. We answer this for k=2, with no Diophantine condition on α: for every irrational α, S_α,2(x)∼ x/ζ(2) as x→∞. More precisely, for every fixed A>0 we obtain an error term O_A(x/(log q(x))^A), where q(x) is the largest continued-fraction convergent denominator of α not exceeding x. The proof sieves S_α,2(x) over divisors, splitting at the parameter R=q(x)^1/8. Divisors larger than R are handled by reducing the condition d|gcd(n,⌊αn^2⌋) to Möbius sums over arithmetic progressions and applying Zhan's uniform estimate for exponential sums twisted by μ(n) over short intervals; divisors at most R are handled instead through the Erdős-Turán inequality together with Weyl-type bounds for quadratic exponential sums of the form ∑ e(βn^2).
ER -