61 research outputs found
A variant of Tao's method with application to restricted sumsets
In this paper, we develop Terence Tao's harmonic analysis method and apply it
to restricted sumsets. The well known Cauchy-Davenport theorem asserts that if
and are nonempty subsets of with a prime, then , where . In 2005, Terence Tao gave
a harmonic analysis proof of the Cauchy-Davenport theorem, by applying a new
form of the uncertainty principle on Fourier transform. We modify Tao's method
so that it can be used to prove the following extension of the Erdos-Heilbronn
conjecture: If are nonempty subsets of with a prime, then
Classification theorems for sumsets modulo a prime
Let be the finite field of prime order and be a subsequence
of . We prove several classification results about the following
questions: (1) When can one represent zero as a sum of some elements of ?
(2) When can one represent every element of as a sum of some elements
of ? (3) When can one represent every element of as a sum of
elements of ?Comment: 35 pages, to appear in JCT
Additive Combinatorics: A Menu of Research Problems
This text contains over three hundred specific open questions on various
topics in additive combinatorics, each placed in context by reviewing all
relevant results. While the primary purpose is to provide an ample supply of
problems for student research, it is hopefully also useful for a wider
audience. It is the author's intention to keep the material current, thus all
feedback and updates are greatly appreciated.Comment: This August 2017 version incorporates feedback and updates from
several colleague
Sumsets with a minimum number of distinct terms
For a non-empty -element set of an additive abelian group and a
positive integer , we consider the set of elements of that can be
written as a sum of elements of with at least distinct elements. We
denote this set as for integers . The set generalizes the classical sumsets and for and
, respectively. Thus, we call the set the generalized
sumset of . By writing the sumset in terms of the sumsets
and , we obtain the sharp lower bound on the size of over the groups and , where is a prime
number. We also characterize the set for which the lower bound on the size
of is tight in these groups. Further, using some elementary
arguments, we prove an upper bound for the minimum size of over
the group for any integer .Comment: 16 page
The number of subsets of integers with no -term arithmetic progression
Addressing a question of Cameron and Erd\Ho s, we show that, for infinitely
many values of , the number of subsets of that do not
contain a -term arithmetic progression is at most , where
is the maximum cardinality of a subset of without
a -term arithmetic progression. This bound is optimal up to a constant
factor in the exponent. For all values of , we prove a weaker bound, which
is nevertheless sufficient to transfer the current best upper bound on
to the sparse random setting. To achieve these bounds, we establish a new
supersaturation result, which roughly states that sets of size
contain superlinearly many -term arithmetic progressions.
For integers and , Erd\Ho s asked whether there is a set of integers
with no -term arithmetic progression, but such that any -coloring
of yields a monochromatic -term arithmetic progression. Ne\v{s}et\v{r}il
and R\"odl, and independently Spencer, answered this question affirmatively. We
show the following density version: for every and , there
exists a reasonably dense subset of primes with no -term arithmetic
progression, yet every of size contains a
-term arithmetic progression.
Our proof uses the hypergraph container method, which has proven to be a very
powerful tool in extremal combinatorics. The idea behind the container method
is to have a small certificate set to describe a large independent set. We give
two further applications in the appendix using this idea.Comment: To appear in International Mathematics Research Notices. This is a
longer version than the journal version, containing two additional minor
applications of the container metho
Arithmetic-Progression-Weighted Subsequence Sums
Let be an abelian group, let be a sequence of terms
not all contained in a coset of a proper subgroup of
, and let be a sequence of consecutive integers. Let
which is a particular kind of weighted restricted sumset. We show that , that if , and also
characterize all sequences of length with . This
result then allows us to characterize when a linear equation
where are
given, has a solution modulo with all
distinct modulo . As a second simple corollary, we also show that there are
maximal length minimal zero-sum sequences over a rank 2 finite abelian group
(where and ) having
distinct terms, for any . Indeed, apart from
a few simple restrictions, any pattern of multiplicities is realizable for such
a maximal length minimal zero-sum sequence
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