On the greatest common divisor of $n, \lfloor \alpha_1n\rfloor, \lfloor \alpha_2n^2\rfloor, ..., \lfloor \alpha_kn^k\rfloor$

Abstract

We answer a question of Bergelson and Richter about the probability of the relation $\gcd(n,\lfloor\alpha_1n\rfloor,\lfloor\alpha_2n^2\rfloor,...,\lfloor\alpha_kn^k\rfloor)=1$ for $n\in\mathbb{N}$ when $\alpha_1,...,\alpha_k$ are fixed irrational numbers. In particular, we avoid the use of exponential sums, as they are difficult to control when one of $\alpha_1,...,\alpha_k$ admits very efficient rational approximations. Instead, we use an elementary method similar to that of Erd\H{o}s and Lorentz, together with some general bounds on those $n$'s which share a large prime divisor with $\lfloor\alpha_1n\rfloor$.