2-large sets are sets of Bohr recurrence
Organizers
Speaker
Sohail Farhangi
Time
Tuesday, December 9, 2025 3:15 PM - 4:15 PM
Venue
A3-1-101
Online
Zoom 482 240 1589
(BIMSA)
Abstract
A set $D \subseteq \mathbb{N}$ is $r$-large if for any partition $\mathbb{N} = \bigcup_{i = 1}^rC_i$, there exists some $1 \le i_0 \le r$ for which $C_{i_0}$ contains arbitrarily long arithmetic progressions whose common difference comes from $D$. If $D$ is $r$-large for all $r \in \mathbb{N}$, then we say that $D$ is large. In 1999, Brown, Graham, and Landman conjectured that if the set $D$ is 2-large, then it is large. In 2016, Farhangi (the speaker) made the weaker conjecture that if $D$ is 2-large, then it is a set of Bohr recurrence. $D \subseteq \mathbb{N}$ is a set of Bohr recurrence if for any real numbers $\alpha_1,\cdots,\alpha_k$ and any $\epsilon > 0$, there exists $d \in D$ with $\|d\alpha_i\|_{\mathbb{T}} < \epsilon$ for all $1 \le i \le k$. We will present the recent work of Alweiss proving Farhangi's conjecture.