26 research outputs found
Matchings in Random Biregular Bipartite Graphs
We study the existence of perfect matchings in suitably chosen induced
subgraphs of random biregular bipartite graphs. We prove a result similar to a
classical theorem of Erdos and Renyi about perfect matchings in random
bipartite graphs. We also present an application to commutative graphs, a class
of graphs that are featured in additive number theory.Comment: 30 pages and 3 figures - Latest version has updated introduction and
bibliograph
Products of Differences over Arbitrary Finite Fields
There exists an absolute constant such that for all and all
subsets of the finite field with elements, if
, then Any suffices for sufficiently large
. This improves the condition , due to Bennett, Hart,
Iosevich, Pakianathan, and Rudnev, that is typical for such questions.
Our proof is based on a qualitatively optimal characterisation of sets for which the number of solutions to the equation is nearly
maximum.
A key ingredient is determining exact algebraic structure of sets for
which is nearly minimum, which refines a result of Bourgain and
Glibichuk using work of Gill, Helfgott, and Tao.
We also prove a stronger statement for when are sets in a prime field,
generalising a result of Roche-Newton, Rudnev, Shkredov, and the authors.Comment: 42 page
A point-line incidence identity in finite fields, and applications
Let be a set in the 2-dimensional vector space
over a finite field with elements. We prove an identity for the second
moment of its incidence function and deduce a variety of existing results from
the literature, not all naturally associated with lines in , in
a unified and elementary way.Comment: 30 page