Incomplete Kloosterman Research Articles

Bounds on bilinear forms with Kloosterman sums

AbstractWe prove new bounds on bilinear forms with Kloosterman sums, complementing and improving a series of results by É. Fouvry, E. Kowalski and Ph. Michel (2014), V. Blomer, É. Fouvry, E. Kowalski, Ph. Michel and D. Milićević (2017), E. Kowalski, Ph. Michel and W. Sawin (2019, 2020) and I. E. Shparlinski (2019). These improvements rely on new estimates for Type II bilinear forms with incomplete Kloosterman sums. We also establish new estimates for bilinear forms with one variable from an arbitrary set by introducing techniques from additive combinatorics over prime fields. Some of these bounds have found a crucial application in the recent work of Wu (2020) on asymptotic formulas for the fourth moments of Dirichlet ‐functions. As new applications, an estimate for higher moments of averages of Kloosterman sums and the distribution of divisor function in a family of arithmetic progressions are also given.

New estimates for short Kloosterman sums with weights

In the paper we obtain a new bound for short Kloosterman sums modulo a prime with a weight. The derivation of the bound is based on Karatsuba's method (1993-1995) of estimating incomplete Kloosterman sums and on a modification of the method proposed by Bourgain and Garaev (2014). The theorems proved in the paper refine results obtained earlier by Korolev (2010).

Суммы Клоостермана по простым числам и разрешимость одного сравнения с обратными вычетами — II

В настоящей статье продолжены исследования, связанные с распределением обратных вычетов по заданному модулю. Ранее автором был получен ряд нетривиальных оценок коротких сумм Клоостермана с простыми числами, отвечающих произвольному модулю $q$. Следствием таких оценок стали результаты о распределении вычетов $\overline{p}$, обратных простым числам "короткого" промежутка: $p\overline{p}\equiv 1\pmod{q}$, $1<p\leqslant N$, $N\leqslant q^{1-\delta}$, $\delta>0$, и, более общо, о распределении по модулю $q$ величин $g(p) = a\overline{p}+bp$, где $a,b$ - целые числа, $(ab,q)=1$.Еще одно приложение найденных оценок связано с задачей о представимости произвольного заданного вычета $m\pmod{q}$ суммою $g(p_{1})+\ldots+g(p_{k})$ при фиксированных $a,b$ и $k\geqslant 3$, и простых $1<p_{1},\ldots,p_{k}\leqslant N$. Для количества таких представлений автором была найдена формула, поведение предполагаемого главного члена которой определяется аналогом "сингулярного ряда" классического кругового метода, т. е. некоторой величиной $\kappa$, зависящей от $q$ и набора $k, a, b, m$. При фиксированных $k, a, b, m$ она является мультипликативной функцией $q$. В случае, когда модуль $q$ не делится на 2 или 3, эта величина строго положительна, так что формула для искомого числа представлений является асимптотической.В настоящей работе исследуется поведение $\kappa$ в случае, когда $q = 3^{n}$. Оказывается, что при любых $n\geqslant 1$, $k\geqslant 3$ существуют "исключительные" тройки $a, b, m$, для которых $\kappa = 0$. Цель работы состоит в описании всех таких троек и нижней оценки величины $\kappa$ для "неисключительных" троек.

Bounds on average values of double incomplete Kloosterman sums

We employ some ideas of D. R. Heath-Brown (1982–1986) to obtain a new bound on certain sums of double incomplete Kloosterman sums, which generalises and improves in some ranges of parameters a previous bound of J. B. Friedlander and H. Iwaniec (1985).

Elementary Proof of an Estimate for Kloosterman Sums with Primes

A new elementary proof of an estimate for incomplete Kloosterman sums modulo a prime q is obtained. Along with Bourgain’s 2005 estimate of the double Kloosterman sum of a special form, it leads to an elementary derivation of an estimate for Kloosterman sums with primes for the case in which the length of the sum is of order q0.5+e, where e is an arbitrarily small fixed number.

МЕТОДЫ ОЦЕНОК КОРОТКИХ СУММ КЛООСТЕРМАНА

This survey contains enlarged version of a mini-course which was read by the author in November 2015 during “Chinese - Russian workshop of exponential sums and sumsets”. This workshop was organized by professors Chaohua Jia (Institute of Mathematics, Academia Sinica) and Ke Gong (Henan University) in Academy of Mathematics and System Science, CAS (Beijing). The author is warmly grateful to them for the support and hospitality. The survey contains the Introduction, three parts and Conclusion. The basic definitions and results concerning the complete Kloosterman sums are given in the Introduction. The method of estimating of incomplete Kloosterman sums to moduli equal to the raising power of a fixed prime is described in the first part. This method is based on one idea of A. G. Postnikov which reduces the estimate of such sums to the estimate of the exponential sums with polynomial by I. M. Vinogradov’s mean value theorem. A. A. Karatsuba’s method of estimating of incomplete sums to an arbitrary moduli is described in the second part. This method is based on a very precise estimate of the number of solutions of one symmetric congruence involving inverse residues to a given modulus. This estimate plays the same role in thie problems under considering as Vinogradov’s mean value theorem in the estimating of corresponding exponential sums. The method of J. Bourgain and M. Z. Garaev is described in the third part. This method is based on very deep “sum-product estimate” and on the improvement of A. A. Karatsuba’s bound for the number of solutions of symmetric congruence. The Conclusion contains a series of recent results concerning the estimates of short Kloosterman sums.

INCOMPLETE KLOOSTERMAN SUMS AND MULTIPLICATIVE INVERSES IN SHORT INTERVALS

We investigate the solubility of the congruence xy ≡ 1 ( mod p), where p is a prime and x, y are restricted to lie in suitable short intervals. Our work relies on a mean value theorem for incomplete Kloosterman sums.

On s-dimensional incomplete Kloosterman sums

In this paper we utilize the estimation of number of solutions of congruence to obtain the upper bound of incomplete Kloosterman sums, which improves the Shparlinski's result and removes the parameter r.

Sets of the form and finite continued fractions

Estimates are obtained for the number of proper irreducible fractions with denominator such that an initial and an end segment of their expansion in a continued fraction have bounded partial quotients. These results are connected with an estimate of incomplete Kloosterman sums over sets of the form . Results on the distribution in of the elements of sets of the form and are obtained. Bibliography: 21 titles.

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

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]).

Incomplete Kloosterman sums and their applications

We obtain a non-trivial estimate for the upper bound of the absolute value of incomplete Kloosterman sums in which the number of terms is much less than the modulus.

Identities of incomplete Kloosterman sums

Identities between incomplete Kloosterman sums and incomplete hyper-Kloosterman sums are established.

On the location of the roots of polynomial congruences

We have indicated in our tract [9] that several interesting problems in the theory of numbers are related to results about the evenness of the distribution of the roots v of a polynomial congruencewhere f(x) = a0xn + … + an is an irreducible polynomial having integral coefficients and degree n≧2. We alluded, for example, to our work on the Chebyshev problem of the greatest prime factor of n2 – D [8], in which an essential component was our earlier demonstration [6] of the uniform distribution, modulo 1, of v/k when f(x) = x2 – D. But, having pointed out that the quantitative descriptions of such uniformity had to be very sharp for substantial applications, we then noted with regret that little more than mere uniform distribution was obtained in our generalization [7] of [6] to congruences of higher degree. Indeed, it has only been for certain cubic polynomials that results have been produced that are comparable in power with those for quadratic polynomials, and even these depend on the assumption of the unproved hypothesis R* regarding the size of incomplete Kloosterman sums [10].

Incomplete Kloosterman Sums and a Divisor Problem

Incomplete Kloosterman Sums and a Divisor Problem