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In this paper we prove some results on sum-product estimates over arbitrary finite fields. More precisely, we show that for sufficiently small sets $Asubset mathbb{F}_q$ we have [|(A-A)^2+(A-A)^2|gg |A|^{1+frac{1}{21}}.] This can be viewed as the ErdH{o}s distinct distances problem for Cartesian product sets over arbitrary finite fields. We also prove that [max{|A+A|, |A^2+A^2|}gg |A|^{1+frac{1}{42}}, ~|A+A^2|gg |A|^{1+frac{1}{84}}.]
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We study some sum-product problems over matrix rings. Firstly, for $A, B, Csubseteq M_n(mathbb{F}_q)$, we have $$ |A+BC|gtrsim q^{n^2}, $$ whenever $|A||B||C|gtrsim q^{3n^2-frac{n+1}{2}}$. Secondly, if a set $A$ in $M_n(mathbb{F}_q)$ satisfies $|A|geq C(n)q^{n^2-1}$ for some sufficiently large $C(n)$, then we have $$ max{|A+A|, |AA|}gtrsim minleft{frac{|A|^2}{q^{n^2-frac{n+1}{4}}}, q^{n^2/3}|A|^{2/3}right}. $$ These improve the results due to The and Vinh (2020), and generalize the results due to Mohammadi, Pham, and Wang (2021). We also give a new proof for a recent result due to The and Vinh (2020). Our method is based on spectral graph theory and linear algebra.
In this paper, we prove some extensions of recent results given by Shkredov and Shparlinski on multiple character sums for some general families of polynomials over prime fields. The energies of polynomials in two and three variables are our main ingredients.
In recent years, sum-product estimates in Euclidean space and finite fields have been studied using a variety of combinatorial, number theoretic and analytic methods. Erdos type problems involving the distribution of distances, areas and volumes have also received much attention. In this paper we prove a relatively straightforward function version of an incidence results for points and planes previously established in cite{HI07} and cite{HIKR07}. As a consequence of our methods, we obtain sharp or near sharp results on the distribution of volumes determined by subsets of vector spaces over finite fields and the associated arithmetic expressions. In particular, our machinery enables us to prove that if $E subset {Bbb F}_q^d$, $d ge 4$, the $d$-dimensional vector space over a finite field ${Bbb F}_q$, of size much greater than $q^{frac{d}{2}}$, and if $E$ is a product set, then the set of volumes of $d$-dimensional parallelepipeds determined by $E$ covers ${Bbb F}_q$. This result is sharp as can be seen by taking $E$ to equal to $A times A times ... times A$, where $A$ is a sub-field of ${Bbb F}_q$ of size $sqrt{q}$. In three dimensions we establish the same result if $|E| gtrsim q^{{15/8}}$. We prove in three dimensions that the set of volumes covers a positive proportion of ${Bbb F}_q$ if $|E| ge Cq^{{3/2}}$. Finally we show that in three dimensions the set of volumes covers a positive proportion of ${Bbb F}_q$ if $|E| ge Cq^2$, without any further assumptions on $E$, which is again sharp as taking $E$ to be a 2-plane through the origin shows.
If the $ell$-adic cohomology of a projective smooth variety, defined over a $frak{p}$-adic field $K$ with finite residue field $k$, is supported in codimension $ge 1$, then any model over the ring of integers of $K$ has a $k$-rational point. This slightly improves our earlier result math/0405318: we needed there the model to be regular (but then our result was more general: we obtained a congruence for the number of points, and $K$ could be local of characteristic $p>0$).
If the $ell$-adic cohomology of a projective smooth variety, defined over a local field $K$ with finite residue field $k$, is supported in codimension $ge 1$, then every model over the ring of integers of $K$ has a $k$-rational point. For $K$ a $p$-adic field, this is math/0405318, Theorem 1.1. If the model $sX$ is regular, one has a congruence $|sX(k)|equiv 1 $ modulo $|k|$ for the number of $k$-rational points 0704.1273, Theorem 1.1. The congruence is violated if one drops the regularity assumption.
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