Pinned distance sets, k-simplices, Wolff’s exponent in finite fields and sum-product estimates

Abstract

Abstract. An analog of the Falconer distance problem in vector spaces over finite fields asks for the threshold α > 0 such that |∆(E)| q whenever |E| q α , where E ⊂ F d q , the d-dimensional vector space over a finite field with q elements (not necessarily prime). Here x ∈ E} for a pin y ∈ E has been studied in the Euclidean setting. Peres and Schlag ([25]) showed that if the Hausdorff dimension of a set E is greater than d+1 2 then the Lebesgue measure of ∆y(E) is positive for almost every pin y. In this paper we obtain the analogous result in the finite field setting. In addition, the same result is shown to be true for the pinned dot product set Πy(E) = {x · y : x ∈ E}. Under the additional assumption that the set E has cartesian product structure we improve the pinned threshold for both distances and dot products to

Product graphs, sum-product graphs and sum-product estimates over finite rings

“…We actually can prove a stronger form of this theorem. More precisely, the author ( [14]) gave a graph-theoretic proof of the following result (see also [3,13] for analogous results in the finite field setting). …”

Long Paths in the Distance Graph Over Large Subsets of Vector Spaces Over Finite Fields

“…For more details on this result and for other results on the distance graph of F d q , see [1], [3], [6] and [7].…”

Additive Combinatorics over Finite Fields: New Results and Applications

“…, see also [64,201,205]. Several more interesting results in this direction have been given by Hegyvári [121,122] and Vinh [198,201,205,206].…”

“…Furthermore, Schoen & Shkredov [167] have successfully used a "cubic" generalization of the energy. We also have had to leave out such exciting areas of additive combinatorics in finite fields as • the Erdős distance problem [83,94,117,130,132,144,145] as well as its modification in some other settings (distinct volumes, configurations, and so on defined by arbitrary sets in F n q ) and metrics [14,64,142,195,200,202,203,205]; • the Kakeya problem and other related problems about the directions defined by arbitrary sets in vector spaces over a finite field, see [84-86, 88, 128, 131, 151]; • estimating the size of the sets in a finite field that avoid arithmetic or geometric progressions, sum sets and similar linear and non-linear relations; in particular these results include finite field analogues of the Roth and Szemerédi theorems, see [1,6,8,12,77,81,112,113,153,154,158,181]; • estimating the size of the sets in vector spaces over a finite field that define only some restrictive geometric configurations such as integral distances, acute angles, and pairwise orthogonal systems, see [75,133,134,183,194,204]; • distribution of the values of determinants and permanents of matrices with entries from general sets, see [74,196,197]; and several others.…”