178 research outputs found
A superadditivity and submultiplicativity property for cardinalities of sumsets
For finite sets of integers A1, . . . ,An we study the cardinality of the n-fold
sumset A1 + · · · + An compared to those of (n − 1)-fold sumsets A1 + · · · + Ai−1 +
Ai+1 + · · · + An. We prove a superadditivity and a submultiplicativity property for
these quantities. We also examine the case when the addition of elements is restricted
to an addition graph between the sets
Sumset and Inverse Sumset Inequalities for Differential Entropy and Mutual Information
The sumset and inverse sumset theories of Freiman, Pl\"{u}nnecke and Ruzsa,
give bounds connecting the cardinality of the sumset of two discrete sets , to the cardinalities (or the finer
structure) of the original sets . For example, the sum-difference bound of
Ruzsa states that, , where the difference set . Interpreting the differential entropy of a
continuous random variable as (the logarithm of) the size of the effective
support of , the main contribution of this paper is a series of natural
information-theoretic analogs for these results. For example, the Ruzsa
sum-difference bound becomes the new inequality, , for any pair of independent continuous random variables and .
Our results include differential-entropy versions of Ruzsa's triangle
inequality, the Pl\"{u}nnecke-Ruzsa inequality, and the
Balog-Szemer\'{e}di-Gowers lemma. Also we give a differential entropy version
of the Freiman-Green-Ruzsa inverse-sumset theorem, which can be seen as a
quantitative converse to the entropy power inequality. Versions of most of
these results for the discrete entropy were recently proved by Tao,
relying heavily on a strong, functional form of the submodularity property of
. Since differential entropy is {\em not} functionally submodular, in the
continuous case many of the corresponding discrete proofs fail, in many cases
requiring substantially new proof strategies. We find that the basic property
that naturally replaces the discrete functional submodularity, is the data
processing property of mutual information.Comment: 23 page
Sumsets and entropy revisited
The entropic doubling of a random variable
taking values in an abelian group is a variant of the notion of the
doubling constant of a finite subset of , but it enjoys
somewhat better properties; for instance, it contracts upon applying a
homomorphism.
In this paper we develop further the theory of entropic doubling and give
various applications, including:
(1) A new proof of a result of P\'alv\"olgyi and Zhelezov on the ``skew
dimension'' of subsets of with small doubling;
(2) A new proof, and an improvement, of a result of the second author on the
dimension of subsets of with small doubling;
(3) A proof that the Polynomial Freiman--Ruzsa conjecture over
implies the (weak) Polynomial Freiman--Ruzsa conjecture over .Comment: 37 page
Self similar sets, entropy and additive combinatorics
This article is an exposition of recent results on self-similar sets,
asserting that if the dimension is smaller than the trivial upper bound then
there are almost overlaps between cylinders. We give a heuristic derivation of
the theorem using elementary arguments about covering numbers. We also give a
short introduction to additive combinatorics, focusing on inverse theorems,
which play a pivotal role in the proof. Our elementary approach avoids many of
the technicalities in the original proof but also falls short of a complete
proof. In the last section we discuss how the heuristic argument is turned into
a rigorous one.Comment: 21 pages, 2 figures; submitted to Proceedings of AFRT 2012. v5: more
typos correcte
Sampling-based proofs of almost-periodicity results and algorithmic applications
We give new combinatorial proofs of known almost-periodicity results for
sumsets of sets with small doubling in the spirit of Croot and Sisask, whose
almost-periodicity lemma has had far-reaching implications in additive
combinatorics. We provide an alternative (and L^p-norm free) point of view,
which allows for proofs to easily be converted to probabilistic algorithms that
decide membership in almost-periodic sumsets of dense subsets of F_2^n.
As an application, we give a new algorithmic version of the quasipolynomial
Bogolyubov-Ruzsa lemma recently proved by Sanders. Together with the results by
the last two authors, this implies an algorithmic version of the quadratic
Goldreich-Levin theorem in which the number of terms in the quadratic Fourier
decomposition of a given function is quasipolynomial in the error parameter,
compared with an exponential dependence previously proved by the authors. It
also improves the running time of the algorithm to have quasipolynomial
dependence instead of an exponential one.
We also give an application to the problem of finding large subspaces in
sumsets of dense sets. Green showed that the sumset of a dense subset of F_2^n
contains a large subspace. Using Fourier analytic methods, Sanders proved that
such a subspace must have dimension bounded below by a constant times the
density times n. We provide an alternative (and L^p norm-free) proof of a
comparable bound, which is analogous to a recent result of Croot, Laba and
Sisask in the integers.Comment: 28 page
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