Feb. 22, 3:30pm, Wean 8220

Yufei Zhao, MIT

Tower-type bounds for Roth's theorem with popular differences

Yufei Zhao, MIT

Tower-type bounds for Roth's theorem with popular differences

Abstract:

A famous theorem of Roth states that for any *α* > *0* and *n*
sufficiently large in terms of *α*, any subset of {*1*, ..., *n*}
with density *α* contains a *3*-term arithmetic progression. Green
developed an arithmetic regularity lemma and used it to prove that not only
is there one arithmetic progression, but in fact there is some integer *d* >
*0* for which the density of *3*-term arithmetic progressions with common
difference *d* is at least roughly what is expected in a random set with
density *α*. That is, for every *ε* > *0*, there is some
*n*(ε) such that for all *n* > *n*(*ε*) and any subset *A* of
{*1*, ..., *n*} with density α, there is some integer *d* > *0* for
which the number of *3*-term arithmetic progressions in *A* with common
difference *d* is at least (*α ^{3}*-

Joint work with Jacob Fox and Huy Tuan Pham.

Before the talk, at 3:10pm, there will be tea and cookies in Wean 6220.