scholarly journals Estimations on some hybrid exponential sums related to Kloosterman sums

2021 ◽  
Keyword(s):  
Exponential Sums ◽  
Kloosterman Sums

scholarly journals Kloosterman paths and the shape of exponential sums

2016 ◽  
Vol 152 (7) ◽  
pp. 1489-1516 ◽  
Author(s):  
Emmanuel Kowalski ◽  
William F. Sawin
Keyword(s):  
Fourier Series ◽  
Exponential Sums ◽  
Continuous Functions ◽  
Kloosterman Sums ◽  
Partial Sums ◽  
Space Of Continuous Functions ◽  
Convergence In Law ◽  
Random Fourier Series ◽  
Polygonal Paths

We consider the distribution of the polygonal paths joining partial sums of classical Kloosterman sums$\text{Kl}_{p}(a)$, as$a$varies over$\mathbf{F}_{p}^{\times }$and as$p$tends to infinity. Using independence of Kloosterman sheaves, we prove convergence in the sense of finite distributions to a specific random Fourier series. We also consider Birch sums, for which we can establish convergence in law in the space of continuous functions. We then derive some applications.

MORE ON THE SUM-PRODUCT PHENOMENON IN PRIME FIELDS AND ITS APPLICATIONS

2005 ◽  
Vol 01 (01) ◽  
pp. 1-32 ◽  
Author(s):  
J. BOURGAIN
Keyword(s):  
Finite Fields ◽  
Exponential Sums ◽  
Rational Functions ◽  
Exponential Sum ◽  
Kloosterman Sums ◽  
Product Theorem ◽  
Short Intervals ◽  
Prime Fields ◽  
Product Sets ◽  
Mod P

In this paper we establish new estimates on sum-product sets and certain exponential sums in finite fields of prime order. Our first result is an extension of the sum-product theorem from [8] when sets of different sizes are involed. It is shown that if [Formula: see text] and pε < |B|, |C| < |A| < p1-ε, then |A + B| + |A · C| > pδ (ε)|A|. Next we exploit the Szemerédi–Trotter theorem in finite fields (also obtained in [8]) to derive several new facts on expanders and extractors. It is shown for instance that the function f(x,y) = x(x+y) from [Formula: see text] to [Formula: see text] satisfies |F(A,B)| > pβ for some β = β (α) > α whenever [Formula: see text] and |A| ~ |B|~ pα, 0 < α < 1. The exponential sum ∑x∈ A,y∈Bεp(axy+bx2y2), ab ≠ 0 ( mod p), may be estimated nontrivially for arbitrary sets [Formula: see text] satisfying |A|, |B| > pρ where ρ < 1/2 is some constant. From this, one obtains an explicit 2-source extractor (with exponential uniform distribution) if both sources have entropy ratio at last ρ. No such examples when ρ < 1/2 seemed known. These questions were largely motivated by recent works on pseudo-randomness such as [2] and [3]. Finally it is shown that if pε < |A| < p1-ε, then always |A + A|+|A-1 + A-1| > pδ(ε)|A|. This is the finite fields version of a problem considered in [11]. If A is an interval, there is a relation to estimates on incomplete Kloosterman sums. In the Appendix, we obtain an apparently new bound on bilinear Kloosterman sums over relatively short intervals (without the restrictions of Karatsuba's result [14]) which is of relevance to problems involving the divisor function (see [1]) and the distribution ( mod p) of certain rational functions on the primes (cf. [12]).

scholarly journals Fractional parts of Dedekind sums

2016 ◽  
Vol 12 (05) ◽  
pp. 1137-1147
Author(s):  
William D. Banks ◽  
Igor E. Shparlinski
Keyword(s):  
Number Theory ◽  
Uniform Distribution ◽  
Exponential Sums ◽  
Kloosterman Sums ◽  
Duke Math ◽  
Dedekind Sums ◽  
Bilinear Forms ◽  
Recent Improvement ◽  
Double Exponential ◽  
Invent Math

Using a recent improvement by Bettin and Chandee to a bound of Duke, Friedlander and Iwaniec [Bilinear forms with Kloosterman fractions, Invent. Math. 128 (1997) 23–43] on double exponential sums with Kloosterman fractions, we establish a uniformity of distribution result for the fractional parts of Dedekind sums [Formula: see text] with [Formula: see text] and [Formula: see text] running over rather general sets. Our result extends earlier work of Myerson [Dedekind sums and uniform distribution, J. Number Theory 28 (1988) 233–239] and Vardi [A relation between Dedekind sums and Kloosterman sums, Duke Math. J. 55 (1987) 189–197]. Using different techniques, we also study the least denominator of the collection of Dedekind sums [Formula: see text] on average for [Formula: see text].

scholarly journals Exponential Sums Over Finite Fields and the Large Sieve

2018 ◽  
Vol 2020 (20) ◽  
pp. 7139-7174 ◽  
Author(s):  
Corentin Perret-Gentil
Keyword(s):  
Finite Fields ◽  
Exponential Sums ◽  
Kloosterman Sums ◽  
Large Sieve ◽  
Density Estimates ◽  
Zero Density ◽  
Monodromy Groups ◽  
Cyclotomic Integers

Abstract By using a variant of the large sieve for Frobenius in compatible systems developed in [24] and [27], we obtain zero-density estimates for arguments of $\ell $-adic trace functions over finite fields with values in some algebraic subsets of the cyclotomic integers, when the monodromy groups are known. This applies in particular to hyper-Kloosterman sums and general exponential sums considered by Katz.

scholarly journals Infinite families of recursive formulas generating power moments of Kloosterman sums: O −(2n,2r) case

2013 ◽  
Vol 63 (4) ◽  
Author(s):  
Dae Kim
Keyword(s):  
Exponential Sums ◽  
Linear Codes ◽  
Kloosterman Sums ◽  
Orthogonal Groups ◽  
Power Moment ◽  
Binary Linear Codes ◽  
Maximal Parabolic Subgroup ◽  
Double Cosets ◽  
Recursive Formulas ◽  
Power Moments

AbstractIn this paper, we construct eight infinite families of binary linear codes associated with double cosets with respect to a certain maximal parabolic subgroup of the special orthogonal group SO −(2n, 2r). And we obtain four infinite families of recursive formulas for the power moments of Kloosterman sums and four those of 2-dimensional Kloosterman sums in terms of the frequencies of weights in the codes. This is done via Pless power moment identity and by utilizing the explicit expressions of exponential sums over those double cosets related to the evaluations of “Gauss sums” for the orthogonal groups O −(2n, 2r).

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