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SUMMARY:Solving linear equations in additive sets - Pablo Candela (Univers
 ity of Cambridge)
DTSTART:20101123T160000Z
DTEND:20101123T170000Z
UID:TALK26182@talks.cam.ac.uk
CONTACT:Tom Sanders
DESCRIPTION:Given an affine-linear form L in t variables with integer coef
 ficients\, a subset A of [N]={1\,2\,...\,N} is said to be L-free if A^t do
 es not contain any (non-trivial) solution of the equation L(x)=0. The grea
 test cardinality that an L-free subset of [N] can have is denoted r_L(N).\
 n\nI will discuss recent joint work with Olof Sisask which proves the conv
 ergence of r_L(N)/N (and of other related quantities) as N tends to infini
 ty\, for any given form L in at least 3 variables. The proof uses the disc
 rete Fourier transform and tools from arithmetic combinatorics. The conver
 gence result addresses a question of Imre Ruzsa and extends work of Ernie 
 Croot.\n\nIn the different context where intervals [N] are replaced by cyc
 lic groups of prime order\, we have similar convergence results\, and I wi
 ll discuss how in this context the limits can be related to natural analog
 ous quantities defined on the circle group.
LOCATION:MR4\, CMS
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