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UID:www.tcs.tifr.res.in/event/1316
DTSTAMP:20230914T125959Z
SUMMARY:Strong bounds for three-term progressions
DESCRIPTION:Speaker: Raghu Meka (University of California\, Los Angeles)\n\
nAbstract: \nSuppose you have a set $S$ of integers from $\\{1\,2\,...\,N\
\}$ that contains at least $N / C$ elements. Does such a set contain three
equally spaced numbers (i.e.\, a 3-term arithmetic progression)? For exam
ple\, the set $S$ from $\\{1\,2\,...\, 40\\}$ comprised of the following
15 numbers $\\{1\,2\,4\,5\,10\,11\,13\,14\,28\,29\,31\,32\,37\,38\,40\\}$
avoids every 3-term arithmetic progression.\nWhat is the largest value of
$C$ such that for large enough $N$\, every such set $S$ necessarily conta
ins a 3-term arithmetic progression?\nIn 1953\, Roth showed this is the ca
se when $C$ is roughly (log log N). Behrend in 1946 showed that $C$ can be
at most $exp(\\sqrt(\\log N))$. Since then\, the problem has been a corne
rstone of the area of additive combinatorics. Following a series of remark
able results\, a celebrated paper from 2020 due to Bloom and Sisask improv
ed the lower bound on $C$ to $C = (\\log N)^{1+c}$ for some constant $c >
0$.\nThis talk will describe a new work showing that $C$ can be as big as
$exp((log N)^{0.08})$\, thus getting closer to Behrend's construction. Bas
ed on joint work with Zander Kelley (Univ of Illinois\, Urbana Champaign).
\n
URL:https://www.tcs.tifr.res.in/web/events/1316
DTSTART;TZID=Asia/Kolkata:20230719T160000
DTEND;TZID=Asia/Kolkata:20230719T170000
LOCATION:AG-66
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