2024 Bounds for exponential sums

Bounds for exponential sums

Author: jgpl

August undefined, 2024

Webbounds 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® ... WebMarch 1957 Bounds for exponential sums L. Carlitz , S. Uchiyama Duke Math. J. 24 (1): 37-41 (March 1957). DOI: 10.1215/S0012-7094-57-02406-7 ABOUT FIRST PAGE …

Estimates on polynomial exponential sums SpringerLink

WebMar 26, 2024 · The exponential sum over the binary field has integer values and very often it also has divisibility properties that permit further improvements. In pursuing these ideas … 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; how far is march 7th

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WebIn this paper we consider exponential sums of the type (1.1) S(axn + bx,pm)= J2 e(axn + bx/pm), ccmodp771 and (1.2) S(axn + bx,x,Pm)= E x(xWaxn + bx/pm), xmodp™ where … WebFeb 1, 2024 · The bound we obtain is always weaker than (1.2) however it applies to more general sums, essentially for arbitrary r and s, where is not necessary an initial interval but can be in an arbitrary position. 2. General notation For complex weights , supported on a set , we define the norms where . 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 … how far is marco island from ft myers fl

Bounds of Trilinear and Trinomial Exponential Sums

Exponential Sums over Finite Fields SpringerLink

WebJul 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). WebAbstract. 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 how far is march 8 from nowWebJun 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 highbit systems ab

"WebWhat 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 … " - Bounds for exponential sums

Bounds for exponential sums

Sharper bounds and new proofs of the exponential function with cotangent

WebSums of sub-exponential random variables Let Xi be independent(⌧ 2 i,bi)-sub-exponential random variables. Then Pn i=1 Xi is (Pn i=1 ⌧ 2 i,b⇤)-sub-exponential, where b⇤ = maxi bi Corollary: If Xi satisfy above, then P 1 n Xn i=1 Xi E[Xi] t! 2exp min (nt2 2 1 n Pn i=1 ⌧ 2 i, nt 2b⇤)!. Prof. John Duchi 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. …

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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 … WebJul 3, 2024 · We also obtain explicit bounds for quadrilinear exponential sums. Download to read the full article text References E. Aksoy Yazici, B. Murphy, M. Rudnev and I. D. …

WebApr 20, 2024 · Consider the following norm of exponenetial sum: I ( N, p, k) = ∫ 0 1 ∫ 0 1 ∑ n = 0 N e 2 π i ( n x + n k y) p d x d y. Bourgain mentioned on Page 118 of. … 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) …

WebDec 1, 2024 · Bounds on multilinear exponential sums We recall the following classical bound of bilinear sums, see, for example, [2, Equation (1.4)] or [10, Lemma 4.1]. Lemma 2.5 For any sets X, Y ⊆ F p and any α = ( α x) x ∈ X, β = ( β y) y ∈ Y, with ∑ x ∈ X α x 2 = A and ∑ y ∈ Y β y 2 = B, we have ∑ x ∈ X ∑ y ∈ Y α x β y e p ( x y) ⩽ p A B. 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 …

WebWe have been looking at deviation inequalities, i.e., bounds on tail probabilities like P(Xn ≥ t)for some statistic Xn. 1. Using moment generating function bounds, for sums of independent r.v.s: Chernoff; Hoeffding; sub-Gaussian, sub-exponential random variables; Bernstein. Today: Johnson-Lindenstrauss. 2. Martingale methods:

WebSub-exponential time Sum-of-Squares lower bounds for Principal Components Analysis. Part of Advances in Neural Information Processing Systems 35 (NeurIPS ... In this work, we study the limits of the powerful Sum of Squares (SoS) family of algorithms for Sparse PCA. SoS algorithms have recently revolutionized robust statistics, leading to ... how far is marco island florida from miamiWebMar 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 ... highbix.comWebFeb 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 … how far is marcy ny from meWebNov 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 … how far is mareeba from port douglasWebMar 19, 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 … highbix bellsWebJul 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 ... highbit technologiesWebPossible 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. high bit systems inc