71 research outputs found
Arithmetic structures in random sets
We extend two well-known results in additive number theory, S\'ark\"ozy's
theorem on square differences in dense sets and a theorem of Green on long
arithmetic progressions in sumsets, to subsets of random sets of asymptotic
density 0. Our proofs rely on a restriction-type Fourier analytic argument of
Green and Green-Tao.Comment: 22 page
Additive structures in sumsets
Suppose that A is a subset of the integers {1,...,N} of density a. We provide
a new proof of a result of Green which shows that A+A contains an arithmetic
progression of length exp(ca(log N)^{1/2}) for some absolute c>0. Furthermore
we improve the length of progression guaranteed in higher sumsets; for example
we show that A+A+A contains a progression of length roughly N^{ca} improving on
the previous best of N^{ca^{2+\epsilon}}.Comment: 28 pp. Corrected typos. Updated references
On Roth's theorem on progressions
We show that if A is a subset of {1,...,N} contains no non-trivial three-term
arithmetic progressions then |A|=O(N/ log^{1-o(1)} N). The approach is somewhat
different from that used in arXiv:1007.5444.Comment: 16 pp. Corrected the proof of the Croot-Sisask Lemma. Corrected
typos. Updated reference
On Sums of Sets of Primes with Positive Relative Density
In this paper we show that if is a subset of the primes with positive
relative density , then must have positive upper density
in
. Our argument applies the techniques developed by Green and
Green-Tao used to find arithmetic progressions in the primes, in combination
with a result on sums of subsets of the multiplicative subgroup of the integers
modulo .Comment: 21 pages, to appear in J. London Math. Soc., short remark added and
typos fixe
Roth's theorem in the primes
We show that any set containing a positive proportion of the primes contains
a 3-term arithmetic progression. An important ingredient is a proof that the
primes enjoy the so-called Hardy-Littlewood majorant property. We derive this
by giving a new proof of a rather more general result of Bourgain which,
because of a close analogy with a classical argument of Tomas and Stein from
Euclidean harmonic analysis, might be called a restriction theorem for the
primes.Comment: 23 pages. Updated references and made some minor changes recommended
by the referee. To appear in Annals of Mathematic
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