WebPossible applications include (but are not limited to) complexity theory, random number generation, cryptography, and coding theory. The main method discussed is based on bounds of exponential sums. Accordingly, the book contains many estimates of such sums, including new estimates of classical Gaussian sums. Webtimes, and to satisfy certain bounds on its derivatives. Under suitable conditions, we can expect cancellations in the sum P n2I e(f(n)). The object of the theory of exponential sums is to detect such cancellations, i.e. to bound the said sum non-trivially. By the triangle inequality, the trivial bound is X n2I e(f(n)) ˝jIj;
Bounds of Trilinear and Quadrilinear Exponential Sums
WebWe also prove a refined estimate on the number of collinear triples in a Cartesian product of multiplicative subgroups and derive stronger bounds for trilinear sums with all variables in multiplicative subgroups. Keywords trilinear exponential sums additive combinatorics eigenvalue method Previous article Next article back WebAug 10, 2024 · This is an expository paper on algebraic aspects of exponential sums over finite fields. This is a new direction. Various examples, results and open problems are … difficulty with numbers
Bounds of Trilinear and Trinomial Exponential Sums
WebFeb 1, 2024 · Here we propose a new approach to bounding such sums. The bound we obtain is always weaker than (1.2) however it applies to more general sums, essentially … WebNov 27, 2024 · Estimating exponential sums is a highlight in analytic number theory. This chapter investigates the main techniques to derive some non-trivial bounds for such sums, such as van der Corput’s methods, Weyl’s schift and Vinogradov’s process. More recent techniques, such as Vaughan’s identity, are treated in detail and non-trivial examples ... WebMar 20, 2015 · 1. Let τ ( n) be the divisor function. Let a be either a constant, or a function of X that is slowly varying with X, say X / log ( X) < a ( X) < X log ( X), for example. I want to lower bound sums of the following form. ∑ 1 ≤ n ≤ X a 1 − τ ( n) D, ( 1) and. ∑ 1 ≤ n ≤ X: n ∈ I a 1 − τ ( n) D, ( 2) where I is an index set of ... difficulty with self care