2-large sets are sets of Bohr recurrence

2025/12/01 by Alweiss, Ryan
#Combinatorics (math.CO) #Dynamical Systems (math.DS) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2512.01997

Abstract

Let α₁, ⋯, α_d be real numbers, and let S be the set of integers s so that ||α_i s||_ℝ/ℤ>δ for some i and some fixed δ>0. We prove S is not \enquote2-large, i.e. there is a 2-coloring of ℕ that avoids arbitrarily long arithmetic progressions with common differences in S.

Citations

• New Obstacles to Multiple Recurrence
• Variations on topological recurrence
• Nil Bohr-sets and almost automorphy of higher order
• On the Set of Common Differences in van der Waerden’s Theorem on Arithmetic Progressions

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• New Obstacles to Multiple Recurrence
• On Alweiss's example for multiple recurrence
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• Bohr sets in triple products of large sets in amenable groups