On the greatest common divisor of $n,\lfloor {\alpha }_{1}n\rfloor ,\lfloor {\alpha }_2{n}^2\rfloor ,...,\lfloor {\alpha }_k{n}^k\rfloor$

Abstract

We answer a question of Bergelson and Richter about the probability of the relation $\gcd (n,\lfloor {\alpha }_{1}n\rfloor ,\lfloor {\alpha }_2{n}^2\rfloor ,...,\lfloor {\alpha }_k{n}^k\rfloor )=1$ 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 Erdős and Lorentz, together with some general bounds on those $n$'s which share a large prime divisor with $\lfloor {\alpha }_{1}n\rfloor$.

12 pages