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CATEGORIES:Isaac Newton Institute Seminar Series
SUMMARY:Marton's Polynomial Freiman-Ruzsa conjecture - Ter
ence Tao (University of California\, Los Angeles)
DTSTART;TZID=Europe/London:20240412T153000
DTEND;TZID=Europe/London:20240412T163000
UID:TALK214051AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/214051
DESCRIPTION:The Freiman-Ruzsa theorem asserts that if a finite
subset $A$ of an $m$-torsion group $G$ is of doub
ling at most $K$ in the sense that $|A+A| \\leq K|
A|$\, then $A$ is covered by at most $m^{K^4+1} K^
2$ cosets of a subgroup $H$ of $G$ of cardinality
at most $|A|$. \; Marton's Polynomial Freiman-
Ruzsa conjecture asserted (in the $m=2$ case\, at
least) that the constant $m^{K^4+1} K^2$ could be
replaced by a polynomial in $K$. \; In joint w
ork with Timothy Gowers\, Ben Green\, and Freddie
Manners\, we establish this conjecture for $m=2$ w
ith a bound of $2K^{12}$ (later improved to $2K^{1
1}$ by Jyun-Jie Liao by a modification of the meth
od)\, and for arbitrary $m$ with a bound of $(2K)^
{O(m^3 \\log m)}$. \; Our proof proceeds by pa
ssing to an entropy-theoretic version of the probl
em\, and then performing an iterative process to r
educe a certain modification of the entropic doubl
ing constant\, taking advantage of the bounded tor
sion to obtain a contradiction when the (modified)
doubling constant is non-trivial but cannot be si
gnificantly reduced. \; By known implications\
, this result also provides polynomial bounds for
inverse theorems for the $U^3$ Gowers uniformity n
orm\, or for linearization of approximate homomorp
hisms.\nIn a collaborative project with Yael Dilli
es and many other contributors\, the proof of the
$m=2$ result has been completely formalized in the
proof assistant language Lean. \; In this tal
k we will present both the original human-readable
proof\, and the process of formalizing it into Le
an.
LOCATION:Seminar Room 1\, Newton Institute
CONTACT:
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