414 research outputs found
An improved sum-product inequality in fields of prime order
This note improves the best known exponent 1/12 in the prime field
sum-product inequality (for small sets) to 1/11, modulo a logarithmic factor.Comment: I spotted that the existing version was, in fact, not the final on
Counting sets with small sumset and applications
We study the number of -element sets with for some (fixed) . Improving results of the first author
and of Alon, Balogh, Samotij and the second author, we determine this number up
to a factor of for most and . As a consequence of
this and a further new result concerning the number of sets with , we deduce that the random
Cayley graph on with edge density~ has no
clique or independent set of size greater than ,
asymptotically the same as for the Erd\H{o}s-R\'enyi random graph. This
improves a result of the first author from 2003 in which a bound of was obtained. As a second application, we show that if the elements of are chosen at random, each with probability , then the
probability that misses exactly elements of is equal to
as .Comment: 30 pages, to appear in Combinatorica. Minor changes made following
helpful suggestions by the referee
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