On integers n relatively prime to αn²

For α and k, let Sα,k(x):=#∈ N:n≤ x,\ (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(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).