On integers $n$ relatively prime to $\lfloor αn^2\rfloor$

Abstract

For $α\in\mathbb{R}$ and $k\in\mathbb{N}$, let $S_{α,k}(x):=\#\{n\in\mathbb N:n\le x,\ \gcd(n,\lfloorαn^k\rfloor)=1\}$. Roth conjectured, and Watson proved, that $S_{α,1}(x)\sim 6x/π^2$ for every irrational $α$. Whether the same asymptotic holds for every fixed $k\ge2$ 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 $α$, $S_{α,2}(x)\sim x/ζ(2)$ as $x\to\infty$. 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 $d\mid\gcd(n,\lfloorαn^2\rfloor)$ 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 $\sum e(βn^2)$.