On the greatest common divisor of n, \lfloor α₁n\rfloor, \lfloor α₂n²\rfloor, ..., \lfloor α_kn^k\rfloor

2026/07/30 by Jérémy Champagne
Mathematics · #math.NT #msc:11N25 #msc:11K06

paper · pdf

12 pages

arxiv created 2026/07/30 · arxiv updated 2026/07/31

Abstract

We answer a question of Bergelson and Richter about the probability of the relation gcd(n,\lfloorα₁n\rfloor,\lfloorα₂n²\rfloor,...,\lfloorα_kn^k\rfloor)=1 for n∈ℕ when α₁,...,α_k are fixed irrational numbers. In particular, we avoid the use of exponential sums, as they are difficult to control when one of α₁,...,α_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 \lfloorα₁n\rfloor.

Citations

• On the greatest common divisor of integer parts of polynomials
• Uniform distribution and Hardy fields

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