site stats

Bounds for exponential sums

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 https://rendez-vu.net

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

Bounds for the tornheim double zeta function — 東京理科大学

Category:BOUNDS ON EXPONENTIAL SUMS AND THE POLYNOMIAL …

Tags:Bounds for exponential sums

Bounds for exponential sums

Bounds for exponential sums - Project Euclid

WebJun 8, 2010 · Abstract In this paper we establish new bounds on exponential sums of high degree for general composite moduli. The sums considered are either Gauss sums or ‘sparse’ and we rely on earlier work in the case of prime modulus. Download to read the full article text References Web22 hours ago · Computational optimal transport (OT) has recently emerged as a powerful framework with applications in various fields. In this paper we focus on a relaxation of the original OT problem, the entropic OT problem, which allows to implement efficient and practical algorithmic solutions, even in high dimensional settings. This formulation, also …

Bounds for exponential sums

Did you know?

WebMar 25, 2003 · Estimates are given for the exponential sum $\sum_ {x=1}^p \exp (2\pi i f (x)/p)$, $p$ a prime and $f$ a nonzero integer polynomial, of interest in cases where the … WebDec 23, 2016 · Estimates are given for the exponential ... s u m ∑ ∑ x = 1 p , exp ( ⋅ 2 π i f ( x) / p), p a prime and f a nonzero integer polynomial, of interest in cases where the Weil …

WebIn the present paper, we give bounds for the Tornheim double zeta function T(s 1, s 2, s 3) when ... 3 +σ 1 &gt; 1−K, where K is a positive integer, from bounds for the Hurwitz zeta function which are shown by Bourgain’s bounds for exponential sums. ...

Webexponential sum S(f,p)= p x=1 e p(f(x)), (1.1) where e p(·)isthe additive character e p(·)=e2πi·/p.Itiswell known that bounds forsuch sums imply corresponding estimates for … WebAbstract. We prove, for a sufficiently small subset A of a prime residue field, an estimate on the number of solutions to the equation ( a 1 − a 2) ( a 3 − a 4) = ( a 5 − a 6) ( a 7 − a 8) …

WebThe boundary of K − 1 ( A) is ∂ K − 1 ( A) = ⋃ m = 1 N − 1 [ K − 1 ( A) ∩ ( { 0 R m } × R N − m)] ⏟ B m. Using the method of Lagrange multipliers, we show that E has a maximum on …

WebDec 1, 2024 · Here we provide some new bounds on quadrinomial exponential sums using the techniques in [13]. We thus define (1.1) Ψ ( X) = a X k + b X ℓ + c X m + d X n. … formula mass of hydrated sodium sulfateWebNov 1, 2014 · In the paper we obtain some new upper bounds for exponential sums over multiplicative subgroups @C@?F"p^@? having sizes in the range [p^c^"^1,p^c^"^2], where c"1, c"2 are some absolute constants cl... On exponential sums over multiplicative subgroups of medium size Finite Fields and Their Applications Advanced Search … difficulty with social communicationWebbounds for exponential sums of this kind. Under the restriction p { ab, p \ n, and * Received September 11, 2000; accepted for publication October 20, 2000. "'"Department of Mathematics, Kansas State University, Manhattan, KS 66506 (cochrane® math.ksu.edu). * Department of Mathematics, Tsinghua University, Beijing 100084, P.R. China (zzheng® ... difficulty with speech is calledWebAbstract. We give upper bounds for the number of solutions to con-gruences with the Euler function ϕ(n) and with the Carmichael function λ(n). We also give nontrivial bounds for certain exponential sums in-volving ϕ(n). Analogous results can also be obtained for the sum of divisors function and similar arithmetic functions. 1 Introduction formula mass of hydrogen peroxide in amuWebWhat you are asking corresponds to estimating the exponential sum $$\sum_{n\leq N}e(\alpha n^{2})$$ on the Major arcs $$\mathfrak{M}_{q,a}=\left\{ \alpha\in[0,1]\ :\ \left \alpha-\frac{a}{q}\right \leq\frac{1}{N^{1-\eta}}\right\}$$ where $\eta\leq\frac{1}{10}.$ Note that this set is larger than what you are asking for, since I allowed … difficulty with speech medicalWebJul 11, 2024 · This answer will requires that: ∫ c r x d x = E i ( r x ln ( c)) ln ( 1 / r) + C. and the bounds: 1 2 exp ( − x) ln ( 1 + ( 2 / x)) ≤ E 1 ( x) ≤ exp ( − x) ln ( 1 + ( 1 / x)) for x ≥ 0, where: E 1 ( x) = − E i ( − x) Since i ↦ c r i is increasing, we have that: ∑ i = 1 n c r i ≤ ∫ i = 1 n + 1 c r x d x = E i ( r n + 1 ln ... difficulty with speech icd 10 codeWebJul 29, 2015 · We establish exponential bounds for the hypergeometric distribution which include a finite sampling correction factor, but are otherwise analogous to bounds for the binomial distribution due to Leön and Perron (2003) and Talagrand (1994). difficulty with speech