Chun-Yen Shen

A SLIGHT IMPROVEMENT TO GARAEV'S SUM PRODUCT ESTIMATE

max(\A + A\,\AA\)>Ce\A\?-*. In the finite field setting this situation is much more complicated because the main tool, the Szemer?di-Trotter incidence theorem, does not hold in the same generality. It is known, via the work in [BKT], that if A is a subset of Fp, the field of p elements with p prime, and if p6 0, then one has the sum product estimate max(|? + A|,|AA|)>|,4|1+ for some e > 0. This result has found many applications in combinatorial problems and exponential sum estimates (see e.g. [BKT], [BGK], [G2]). Recently, Garaev [Gl] showed that when \A\ < pz, one has the estimate

Fourier analysis and expanding phenomena in finite fields

In this paper the authors study set expansion in finite fields. Fourier analytic proofs are given for several results recently obtained by Solymosi, Vinh and Vu using spectral graph theory. In addition, several generalizations of these results are given. In the case that

Sharp extension theorems and Falconer distance problems for algebraic curves in two dimensional vector spaces over finite fields

A

Extension and averaging operators for finite fields

is a subset of a prime field

A Two Weight Fractional Singular Integral Theorem with Side Conditions, Energy and k -Energy Dispersed

\mathbb F_p

A Good-λ Lemma, two weight T1 theorems without weak boundedness, and a two weight accretive global Tb theorem

of size less than

Matrices Whose Norms Are Determined by Their Actions on Decreasing Sequences

p^{1/2}

Expansion for the product of matrices in groups

it is shown that

A slight improvement to Korenblum's constant

|\{a^2+b:a,b \in A\}|\geq C |A|^{147/146}

Averaging Operators Over Homogeneous Varieties Over Finite Fields

, where