We prove a point-wise and average bound for the number of incidences between points and hyper-planes in vector spaces over finite fields. While our estimates are, in general, sharp, we observe an improvement for product sets and sets contained in a sphere. We use these incidence bounds to obtain significant improvements on the arithmetic problem of covering ${\mathbb F}_q$, the finite field with q elements, by $A \cdot A+... +A \cdot A$, where A is a subset ${\mathbb F}_q$ of sufficiently large size. We also use the incidence machinery we develope and arithmetic constructions to study the Erdos-Falconer distance conjecture in vector spaces over finite fields. We prove that the natural analog of the Euclidean Erdos-Falconer distance conjecture does not hold in this setting due to the influence of the arithmetic. On the positive side, we obtain good exponents for the Erdos -Falconer distance problem for subsets of the unit sphere in $\mathbb F_q^d$ and discuss their sharpness. This results in a reasonably complete description of the Erdos-Falconer distance problem in higher dimensional vector spaces over general finite fields.

Averages over hyperplanes, sum-product theory in vector spaces over finite fields and the Erdos-Falconer distance conjecture

Let G = SL3(Z/pZ), p a prime. Let A be a set of generators of G. Then A grows under the group operation.
To be precise: denote by |S| the number of elements of a finite set S. Assume |A| 0. Then |A · A · A| > |A|1+d, where d > 0 depends only on e.

https://www.ems-ph.org/journals/show_pdf.php?issn=1435-9855&vol=13&iss=3&rank=8

Growth in SL3(ℤ/pℤ)

It is proved that for an arbitrary subgroup and any distinct nonzero elements we have under the condition that , where ,  are some sequences of positive numbers such that as . Furthermore, it is shown that the inequality holds for any subgroup  such that . Bibliography: 25 titles.

On additive shifts of multiplicative subgroups

In the paper we develop the method of higher energies. New upper bounds for the additive energies of convex sets, sets A with small |AA| and |A(A+1)| are obtained. We prove new structural results, including higher sumsets, and develop the notion of dual popular difference sets.

/pdf/some-new-results-on-higher-energies-54h3ijhen9.pdf

Some new results on higher energies

/pdf/ob-additivnykh-sdvigakh-multiplikativnykh-podgrupp-on-2sdb1foaw8.pdf

Об аддитивных сдвигах мультипликативных подгрупп@@@On additive shifts of multiplicative subgroups

Let $F_p$ be the field of a prime order $p$. Then for any positive integer $n>1$, for any $\epsilon>0$, and for any subset $A$ of $F_p$, every element of $F_p$ can be represented as a sum of $N$ elements, each of them is a product of $n$ elements from $A$, where $N$ depends on $n$ and $\espilon$.

Additive properties of product sets in fields of prime order

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Additive properties of product sets in fields of prime order