ADDITIVE PROPERTIES OF MULTIPLICATIVE SUBGROUPS OF Fp

On Sumsets of Convex Sets

“…which has recently found many applications [8,10,11,12,13]. Roughly speaking, we show that if |A − A| is not much bigger than |A| 3/2 then the energy E(A, A − A) is very large.…”

Variations on the Sum-Product Problem

“…In recent years, third moment energy has played an important role in quantitative progress on various problems in arithmetic combinatorics. See for example [13], [16], [19], [20], [21], [22] and [27].…”

Additive Combinatorics over Finite Fields: New Results and Applications

“…We start with a recent result of Shkredov & Vyugin [209]: [167] have given a series of bounds on #(G ± G), see also [173]. Many of this results are based on upper bounds for additive energy, that is 1 + a 2 = a 3 + a 4 : a 1 , a 2 , a 3 , a 4 ∈ G of G. Heath-Brown & Konyagin [120] have proved that for #G p 2/3 , we have…”

“…Results of [167,173] about additive representation of F p * but sums of a few elements of a small subgroup G ⊆ F * p also have a natural interpretation as bounds on g(n, p) for a large n.…”

“…Analogues of these notions have also been defined and studied for two distinct sets, see [22,34,108]. 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 entr...…”