109 research outputs found
On a sumset conjecture of Erd\H{o}s
Erd\H{o}s conjectured that for any set with positive
lower asymptotic density, there are infinite sets
such that . We verify Erd\H{o}s' conjecture in the case that
has Banach density exceeding . As a consequence, we prove
that, for with positive Banach density (a much weaker
assumption than positive lower density), we can find infinite such that is contained in the union of and a translate of
. Both of the aforementioned results are generalized to arbitrary countable
amenable groups. We also provide a positive solution to Erd\H{o}s' conjecture
for subsets of the natural numbers that are pseudorandom.Comment: 17 pages; new version has a slightly different title, some minor
typos are fixed, and the exposition of Lemma 4.6 has been improved. To appear
in the Canadian Journal of Mathematic
Combinatorial and Additive Number Theory Problem Sessions: '09--'19
These notes are a summary of the problem session discussions at various CANT
(Combinatorial and Additive Number Theory Conferences). Currently they include
all years from 2009 through 2019 (inclusive); the goal is to supplement this
file each year. These additions will include the problem session notes from
that year, and occasionally discussions on progress on previous problems. If
you are interested in pursuing any of these problems and want additional
information as to progress, please email the author. See
http://www.theoryofnumbers.com/ for the conference homepage.Comment: Version 3.4, 58 pages, 2 figures added 2019 problems on 5/31/2019,
fixed a few issues from some presenters 6/29/201
Mod-phi convergence I: Normality zones and precise deviations
In this paper, we use the framework of mod- convergence to prove
precise large or moderate deviations for quite general sequences of real valued
random variables , which can be lattice or
non-lattice distributed. We establish precise estimates of the fluctuations
, instead of the usual estimates for the rate of
exponential decay . Our approach provides us with a
systematic way to characterise the normality zone, that is the zone in which
the Gaussian approximation for the tails is still valid. Besides, the residue
function measures the extent to which this approximation fails to hold at the
edge of the normality zone.
The first sections of the article are devoted to a proof of these abstract
results and comparisons with existing results. We then propose new examples
covered by this theory and coming from various areas of mathematics: classical
probability theory, number theory (statistics of additive arithmetic
functions), combinatorics (statistics of random permutations), random matrix
theory (characteristic polynomials of random matrices in compact Lie groups),
graph theory (number of subgraphs in a random Erd\H{o}s-R\'enyi graph), and
non-commutative probability theory (asymptotics of random character values of
symmetric groups). In particular, we complete our theory of precise deviations
by a concrete method of cumulants and dependency graphs, which applies to many
examples of sums of "weakly dependent" random variables. The large number as
well as the variety of examples hint at a universality class for second order
fluctuations.Comment: 103 pages. New (final) version: multiple small improvements ; a new
section on mod-Gaussian convergence coming from the factorization of the
generating function ; the multi-dimensional results have been moved to a
forthcoming paper ; and the introduction has been reworke
Sequences with small subsum sets
AbstractA conjecture of Gao and Leader, recently proved by Sun, states that if X=(xi)i=1n is a sequence of length n in a finite abelian group of exponent n, then either some subsequence of X sums to zero or the set of all sums of subsequences of X has cardinality at least 2n−1. This conjecture turns out to be a simple consequence of a theorem of Olson and White; we investigate generalizations that are not implied by this theorem. In particular, we prove the following result: if X=(xi)i=1n is a sequence of length n, the terms of which generate a finite abelian group of rank at least 3, then either some subsequence of X sums to zero or the set of all sums of subsequences of X has cardinality at least 4n−5
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