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SUMMARY:Hu Tan (University of Chinese Academy of Sciences\, China)
DTSTART:20261119T200000Z
DTEND:20261119T210000Z
DTSTAMP:20261107T163508Z
UID:NYNTS/11
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/NYNTS/11/">A
  proof of Fraenkel’s conjecture</a>\nby Hu Tan (University of Chinese Ac
 ademy of Sciences\, China) as part of New York Number Theory Seminar\n\n\n
 Abstract\nFraenkel’s conjecture asserts that any partition of the intege
 rs into $m \\geq 3$ Beatty sequences with pairwise distinct moduli must ha
 ve densities $2^j/(2^m - 1)$ \nfor $j = 0\, …\, m - 1$. In this talk\, I
  will present a proof whose main ingredient is a density bound independent
  of the number of parts: every such partition contains a sequence of densi
 ty at least 1/3. Building on the discrete Fourier framework developed by G
 raham and O’Bryant\, we establish this bound by combining analytic estim
 ates with exact finite verification using integer and rational arithmetic.
  The bound supplies the missing input for Tijdeman’s deletion method\, a
 llowing us to recover the binary density pattern by induction. I will expl
 ain how the Fourier argument connects with the combinatorial reconstructio
 n and discuss the role of the finite computations. This is joint work with
  Professor Ying Zhang.\n
LOCATION:https://researchseminars.org/talk/NYNTS/11/
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