1,976 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
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
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
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