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Simons Institute Event: Strong Bounds for 3-Progressions March 16, 2023 (03:30 PM PDT - 04:30 PM PDT)
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Location: Calvin Lab Auditorium; Virtual
Speaker(s) Raghu Meka (University of California, Los Angeles)
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Please register to attend in-person or to receive the Zoom webinar link.

Suppose you have a set S of integers from {1 , 2 , … , N} that contains at least N / C elements. Then for large enough N , must S contain three equally spaced numbers (i.e., a 3-term arithmetic progression)?

In 1953, Roth showed that this is indeed the case when C ≈ (log ⁡ log ⁡ N), while Behrend in 1946 showed that C can be at most Ω(√log ⁡N). Since then, the problem has been a cornerstone of the area of additive combinatorics. Following a series of remarkable results, a celebrated paper from 2020 due to Bloom and Sisask improved the lower bound on C to C = (log N) (1 + c), for some constant c > 0.

This talk will describe a new work showing that the same holds when C ≈ 2(log ⁡ N)0.09, thus getting closer to Behrend's construction.

Based on joint work with Zander Kelley.


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