A structure theorem for finite fields

A Structure Theorem for Plesken Lie Algebras over Finite Fields

W. Plesken found a simple but interesting construction of a Lie algebra from a finite group. Cohen and Taylor posed themselves the question of what the Plesken Lie algebra, which is the Lie subalgebra of the group algebra k[G] generated by the elements g − g−1, could be. The result is very fascinating: It turns out that the Lie algebra decomposition of the Plesken Lie algebra into simple Lie al...

Structure of finite wavelet frames over prime fields

‎This article presents a systematic study for structure of finite wavelet frames‎ ‎over prime fields‎. ‎Let $p$ be a positive prime integer and $mathbb{W}_p$‎ ‎be the finite wavelet group over the prime field $mathbb{Z}_p$‎. ‎We study theoretical frame aspects of finite wavelet systems generated by‎ ‎subgroups of the finite wavelet group $mathbb{W}_p$.

The Freiman-Ruzsa Theorem in Finite Fields

Let G be a finite abelian group of torsion r and let A be a subset of G. The Freiman-Ruzsa theorem asserts that if |A + A| ≤ K|A| then A is contained in a coset of a subgroup of G of size at most Kr 4 |A|. It was conjectured by Ruzsa that the subgroup size can be reduced to r for some absolute constant C ≥ 2. This conjecture was verified for r = 2 in a sequence of recent works, which have, in f...

Tate’s Isogeny Theorem for Abelian Varieties over Finite Fields

On a Furstenberg-Katznelson-Weiss type theorem over finite fields

Using Fourier analysis, Covert, Hart, Iosevich and Uriarte-Tuero (2008) showed that if the cardinality of a subset of the 2-dimensional vector space over a finite field with q elements is > ρq2, with q−1/2 ≪ ρ 6 1 then it contains an isometric copy of > cρq3 triangles. In this note, we give a graph theoretic proof of this result.