scholarly journals The Maximal Difference of Different Powers of an Element Modulo n

2021 ◽  
Vol 2021 ◽  
pp. 1-5
Author(s):  
Jinyun Qi ◽  
Zhefeng Xu
Keyword(s):  
Lower Bound ◽  
Exponential Sums

In this paper, we investigate the maximal difference of integer powers of an element modulo n . Let a n denote the integer b with 1 ≤ b ≤ n such that a ≡ b mod   n for any integer a . Using the bounds for exponential sums, we obtain a lower bound of the function H m 1 , m 2 n : = max a m 1 n − a m 2 n : 1 ≤ a ≤ n , a , n = 1 , which gives n − H m 1 , m 2 n = O n 3 / 4 + o 1 .

EXPANSION OF ORBITS OF SOME DYNAMICAL SYSTEMS OVER FINITE FIELDS

2010 ◽  
Vol 82 (2) ◽  
pp. 232-239 ◽  
Author(s):  
JAIME GUTIERREZ ◽  
IGOR E. SHPARLINSKI
Keyword(s):  
Dynamical Systems ◽  
Lower Bound ◽  
Finite Field ◽  
Rational Function ◽  
Finite Fields ◽  
Exponential Sums ◽  
Short Interval ◽  
Additive Combinatorics ◽  
Distribution Of Elements ◽  
Image Position

AbstractGiven a finite field 𝔽p={0,…,p−1} of p elements, where p is a prime, we consider the distribution of elements in the orbits of a transformation ξ↦ψ(ξ) associated with a rational function ψ∈𝔽p(X). We use bounds of exponential sums to show that if N≥p1/2+ε for some fixed ε then no N distinct consecutive elements of such an orbit are contained in any short interval, improving the trivial lower bound N on the length of such intervals. In the case of linear fractional functions we use a different approach, based on some results of additive combinatorics due to Bourgain, that gives a nontrivial lower bound for essentially any admissible value of N.

A lower bound for the L 1 norm of exponential sums

Mathematika ◽  
1974 ◽  
Vol 21 (2) ◽  
pp. 155-159 ◽  
Author(s):  
S. K. Pichorides
Keyword(s):  
Lower Bound ◽  
Exponential Sums

The L1-norm of exponential sums in d

2013 ◽  
Vol 154 (3) ◽  
pp. 381-392 ◽  
Author(s):  
GIORGIS PETRIDIS
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Absolute Constant ◽  
Exponential Sums ◽  
Exponential Sum ◽  
Correct Order ◽  
L1 Norm ◽  
Finite Set ◽  
Order Of Magnitude ◽  
Technical Sense

AbstractLet A be a finite set of integers and FA(x) = ∑a∈A exp(2πiax) be its exponential sum. McGehee, Pigno and Smith and Konyagin have independently proved that ∥FA∥1 ≥ c log|A| for some absolute constant c. The lower bound has the correct order of magnitude and was first conjectured by Littlewood. In this paper we present lower bounds on the L1-norm of exponential sums of sets in the d-dimensional grid d. We show that ∥FA∥1 is considerably larger than log|A| when A ⊂ d has multidimensional structure. We furthermore prove similar lower bounds for sets in , which in a technical sense are multidimensional and discuss their connection to an inverse result on the theorem of McGehee, Pigno and Smith and Konyagin.

scholarly journals p-adic Valuation of Exponential Sums in One Variable Associated to Binomials

2017 ◽  
Vol 12 (1) ◽  
pp. 37-53 ◽  
Author(s):  
Francis N. Castro ◽  
Raúl Figueroa ◽  
Puhua Guan
Keyword(s):  
Lower Bound ◽  
Exponential Sums ◽  
Adic Valuation ◽  
Value Sets

Abstract In this paper we compute the p-adic valuation of exponential sums associated to binomials F(X) = aXd₁ + bXd₂ over Fp. In particular, its p-adic valuation is constant for a, b ∈ F∗p . As a byproduct of our results, we obtain a lower bound for the sizes of value sets of binomials over Fq.

Lower bound of overall exponential sums for polynomialsϕ(x d )

1989 ◽  
Vol 20 (4) ◽  
pp. 279-287
Author(s):  
L. A. Bassalygo ◽  
V. A. Zinov'ev ◽  
S. N. Licyn
Keyword(s):  
Lower Bound ◽  
Exponential Sums

scholarly journals ON THE DISTRIBUTION OF THE MAXIMUM OF CUBIC EXPONENTIAL SUMS

2018 ◽  
Vol 19 (4) ◽  
pp. 1259-1286
Author(s):  
Youness Lamzouri
Keyword(s):  
Algebraic Geometry ◽  
Lower Bound ◽  
Exponential Sums ◽  
Probabilistic Methods ◽  
Upper And Lower Bounds ◽  
Partial Sums ◽  
Log P ◽  
Analysis Techniques ◽  
Distribution Of The Maximum ◽  
Order Of Magnitude

In this paper, we investigate the distribution of the maximum of partial sums of certain cubic exponential sums, commonly known as ‘Birch sums’. Our main theorem gives upper and lower bounds (of nearly the same order of magnitude) for the distribution of large values of this maximum, that hold in a wide uniform range. This improves a recent result of Kowalski and Sawin. The proofs use a blend of probabilistic methods, harmonic analysis techniques, and deep tools from algebraic geometry. The results can also be generalized to other types of $\ell$-adic trace functions. In particular, the lower bound of our result also holds for partial sums of Kloosterman sums. As an application, we show that there exist $x\in [1,p]$ and $a\in \mathbb{F}_{p}^{\times }$ such that $|\sum _{n\leqslant x}\exp (2\unicode[STIX]{x1D70B}i(n^{3}+an)/p)|\geqslant (2/\unicode[STIX]{x1D70B}+o(1))\sqrt{p}\log \log p$. The uniformity of our results suggests that this bound is optimal, up to the value of the constant.

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