313 research outputs found
A short proof of a near-optimal cardinality estimate for the product of a sum set
In this note it is established that, for any finite set of real numbers,
there exist two elements such that
In particular, it follows that . The
latter inequality had in fact already been established in an earlier work of
the author and Rudnev (arXiv:1203.6237), which built upon the recent
developments of Guth and Katz (arXiv:1011.4105) in their work on the Erd\H{o}s
distinct distance problem. Here, we do not use those relatively deep methods,
and instead we need just a single application of the Szemer\'{e}di-Trotter
Theorem. The result is also qualitatively stronger than the corresponding
sum-product estimate from (arXiv:1203.6237), since the set is
defined by only two variables, rather than four. One can view this as a
solution for the pinned distance problem, under an alternative notion of
distance, in the special case when the point set is a direct product . Another advantage of this more elementary approach is that these results
can now be extended for the first time to the case when .Comment: To appear in Proceedings of SoCG 201
Convexity and a sum-product type estimate
In this paper we further study the relationship between convexity and
additive growth, building on the work of Schoen and Shkredov (\cite{SS}) to get
some improvements to earlier results of Elekes, Nathanson and Ruzsa
(\cite{ENR}). In particular, we show that for any finite set
and any strictly convex or concave function ,
and For the latter of
these inequalities, we go on to consider the consequences for a sum-product
type problem
Four-term progression free sets with three-term progressions in all large subsets
This paper is mainly concerned with sets which do not contain four-term
arithmetic progressions, but are still very rich in three term arithmetic
progressions, in the sense that all sufficiently large subsets contain at least
one such progression. We prove that there exists a positive constant and a
set which does not contain a four-term arithmetic
progression, with the property that for every subset with , contains a nontrivial three term arithmetic progression.
We derive this from a more general quantitative Roth-type theorem in random
subsets of , which improves a result of
Kohayakawa-Luczak-R\"odl/Tao-Vu. We also discuss a similar phenomenon over the
integers, where we show that for all , and all sufficiently large
, there exists a four-term progression-free set of size
with the property that for every subset with contains a nontrivial three term
arithmetic progression. Finally, we include another application of our methods,
showing that for sets in or the property of
"having nontrivial three-term progressions in all large subsets" is almost
entirely uncorrelated with the property of "having large additive energy".Comment: minor updates including suggestions from referee
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