Exponential sum estimates over a subgroup in an arbitrary finite field

Additive Combinatorics over Finite Fields: New Results and Applications

“…In particular, the results of [162] improve some of the previous results of Gomez and Winterhof [111] that are based on the Weil bound of exponential sums, see [135]. One can probably get further improvements by using the bound of Bourgain & Glibichuk [34,Lemma 5] that has been presented in Theorem 3.3.…”

“…We also present a beautiful result of Bourgain & Glibichuk [34,Lemma 5] that applies to arbitrary fields:…”

“…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.…”

Tree codes and a conjecture on exponential sums

“…However, this bound becomes nontrivial only when N > p 1/2 log p. Improved bounds were given by Bourgain and Garaev [1] and Konyagin and Shparlinski [15]; see also Bourgain and Glibichuk [2] for complete sums over multiplicative subgroups, i.e., over all the powers of some element g that is not a primitive root, and Kerr [14] for incomplete sums of this type. However, all of these only become nontrivial when N ≥ p α for some α > 0.…”

Breaking the k 2 barrier for explicit RIP matrices

“…The construction in [6] gives ε proportional to c 3 0 , while the construction in this paper gives ε = c 2 0 /2530000. The best known value for c0 is c0 = 1 10430 [8] and this implies ε ≈ 3.6 · 10 −15 in Theorem 1, vs. the value ε ≈ 2 · 10 −22 in [6]. It is possible that c0 can be taken close to 1, and this would give a much better constant ε ≈ 3 · 10 −7 .…”