2,226 research outputs found
Clustered Integer 3SUM via Additive Combinatorics
We present a collection of new results on problems related to 3SUM,
including:
1. The first truly subquadratic algorithm for
1a. computing the (min,+) convolution for monotone increasing
sequences with integer values bounded by ,
1b. solving 3SUM for monotone sets in 2D with integer coordinates
bounded by , and
1c. preprocessing a binary string for histogram indexing (also
called jumbled indexing).
The running time is:
with
randomization, or deterministically. This greatly improves the
previous time bound obtained from Williams'
recent result on all-pairs shortest paths [STOC'14], and answers an open
question raised by several researchers studying the histogram indexing problem.
2. The first algorithm for histogram indexing for any constant alphabet size
that achieves truly subquadratic preprocessing time and truly sublinear query
time.
3. A truly subquadratic algorithm for integer 3SUM in the case when the given
set can be partitioned into clusters each covered by an interval
of length , for any constant .
4. An algorithm to preprocess any set of integers so that subsequently
3SUM on any given subset can be solved in
time.
All these results are obtained by a surprising new technique, based on the
Balog--Szemer\'edi--Gowers Theorem from additive combinatorics
Some new inequalities in additive combinatorics
In the paper we find new inequalities involving the intersections of shifts of some subset from an abelian group. We apply the
inequalities to obtain new upper bounds for the additive energy of
multiplicative subgroups and convex sets and also a series another results on
the connection of the additive energy and so--called higher moments of
convolutions. Besides we prove new theorems on multiplicative subgroups
concerning lower bounds for its doubling constants, sharp lower bound for the
cardinality of sumset of a multiplicative subgroup and its subprogression and
another results.Comment: 39 page
Additive combinatorics methods in associative algebras
We adapt methods coming from additive combinatorics in groups to the study of
linear span in associative unital algebras. In particular, we establish for
these algebras analogues of Diderrich-Kneser's and Hamidoune's theorems on
sumsets and Tao's theorem on sets of small doubling. In passing we classify the
finite-dimensional algebras over infinite fields with finitely many
subalgebras. These algebras play a crucial role in our linear version of
Diderrich-Kneser's theorem. We also explain how the original theorems for
groups we linearize can be easily deduced from our results applied to group
algebras. Finally, we give lower bounds for the Minkowski product of two
subsets in finite monoids by using their associated monoid algebras.Comment: In this second version, we clarify and extend the domain of validity
of Diderrich-Kneser's theorem for associative algebras. We simplify the
proofs and we also add a section on Kneser's and Hamidoune's theorem in
monoi
Some Additive Combinatorics Problems in Matrix Rings
We study the distribution of singular and unimodular matrices in sumsets in
matrix rings over finite fields. We apply these results to estimate the largest
prime divisor of the determinants in sumsets in matrix rings over the integers
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