Residue sums of Dickson polynomials over finite fields
Thomas Brazelton, Joshua Harrington, Matthew Litman, Tony W. H. Wong
备注：
Major revisions: we now treat the more general case of Dickson polynomials over finite fields of odd characteristic instead of Lucas polynomials modulo a prime. 19 pages, comments welcome!
Given a polynomial with integral coefficients, one can inquire about the possible residues it can take in its image modulo a prime $p$. The sum over the distinct residues can sometimes be computed independent of the prime $p$; for example, Gauss showed that the sum over quadratic residues vanishes modulo a prime. In this paper we provide a closed form for the sum over distinct residues in the image of Dickson polynomials of arbitrary degree over finite fields of odd characteristic, and prove a complete characterization of the size of the image set. Our result provides the first non-trivial classification of such a sum for a family of polynomials of unbounded degree.
The aim of this paper is to deal with congruences for Jacobi sums of order $2l^{2}$ over a finite field $\mathbb{F}_{q}, q=p^{r}$, $p^{r}\equiv 1\ (mod \ 2l^{2})$, where $l>3$ and $p$ are primes. Further, we also calculate Jacobi sums $J_{2l^{2}}(1,n)$ of order $2l^{2}$ in terms of Dickson-Hurwitz sums.
For an $[n,k]$ Reed-Solomon code $\mathcal{C}$, it can be shown that any received word $r$ lies a distance at most $n-k$ from $\mathcal{C}$, denoted $d(r,\mathcal{C})\leq n-k$. Any word $r$ meeting the equality is called a deep hole. Guruswami and Vardy (2005) showed that for a specific class of codes, determining whether or not a word is a deep hole is NP-hard. They suggested passingly that it may be easier when the evaluation set of $\mathcal{C}$ is large or structured. Following this idea, we study the case where the evaluation set is the image of a Dickson polynomial, whose values appear w…
We prove upper bounds for sums of Kloosterman sums against general arithmetic weight functions. In particular, we obtain power cancellation in sums of Kloosterman sums over arithmetic progressions, which is of square-root strength in any fixed primitive congruence class up to bounds towards the Ramanujan conjecture.
It is well known that when a prime $p$ is congruent to 1 modulo 4, the sum of the quadratic residues equals the sum of the quadratic nonresidues. In this note we give analogous results for the case where $p$ is congruent to 3 modulo 4.
In the present paper, we generalize some of the results on Kloosterman sums proven in \cite{BG} for prime moduli to general moduli. This requires to establish the corresponding additive properties of the reciprocal set $$ I^{-1}=\{x^{-1}:\quad x\in I\}, $$ where $I$ is an interval in the ring of residue classes modulo a large positive integer. We apply our bounds on multilinear exponential sums to the Brun-Titchmarsh theorem and the estimate of very short Kloosterman sums, hence generalizing our earlier work to the setting of general modulus.
Let p be a prime = 3 (mod 4). A number of elegant number-theoretical properties of the sums T(p) = \sqrt{p}sum_{n=1}^{(p-1)/2} tan(n^2π/p) and C(p) = \sqrt{p}sum_{n=1}^{(p-1)/2} cot(n^2π/p) are proved. For example, T(p) equals p times the excess of the odd quadratic residues over the even ones in the set {1,2,...,p-1}; this number is positive if p = 3 (mod 8) and negative if p = 7 (mod 8). In this revised version the connection of these sums with the class-number h(-p) is also discussed. For example, a very simple formula expressing h(-p) by means of the aforementioned excess is proved. The bi…
The $k$-subset sum problem over finite fields is a classical NP-complete problem.Motivated by coding theory applications, a more complex problem is the higher $m$-th moment $k$-subset sum problem over finite fields. We show that there is a deterministic polynomial time algorithm for the $m$-th moment $k$-subset sum problem over finite fields for each fixed $m$ when the evaluation set is the image set of a monomial or Dickson polynomial of any degree $n$. In the classical case $m=1$, this recovers previous findings.
We generalize and solve the $\roman{mod}\,q$ analogue of a problem of Littlewood and Offord, raised by Vaughan and Wooley, concerning the distribution of the $2^n$ sums of the form $\sum_{i=1}^n\varepsilon_ia_i$, where each $\varepsilon_i$ is $0$ or $1$. For all $q$, $n$, $k$ we determine the maximum, over all reduced residues $a_i$ and all sets $P$ consisting of $k$ arbitrary residues, of the number of these sums that belong to $P$.
Let $n\ge 1, e\ge 1, k\ge 2$ and $c$ be integers. An integer $u$ is called a unit in the ring $\mathbb{Z}_n$ of residue classes modulo $n$ if $\gcd(u, n)=1$. A unit $u$ is called an exceptional unit in the ring $\mathbb{Z}_n$ if $\gcd(1-u,n)=1$. We denote by $\mathcal{N}_{k,c,e}(n)$ the number of solutions $(x_1,...,x_k)$ of the congruence $x_1^e+...+x_k^e\equiv c \pmod n$ with all $x_i$ being exceptional units in the ring $\mathbb{Z}_n$. In 2017, Mollahajiaghaei presented a formula for the number of solutions $(x_1,...,x_k)$ of the congruence $x_1^2+...+x_k^2\equiv c\pmod n$ with all $x_i$ be…
In the paper, we establish a new estimate for Kloosterman sum over primes with respect to an arbitrary modulus $q$. This estimate together with some recent results of the second author are applied to the problem of solvability of the congruence \[ g(p_{1})\, + \,\ldots\, + \,g(p_{k}) \,\equiv\, m\pmod{q} \] in prime variables $p_{1},\ldots, p_{k}\le N$, $N\le q^{1-δ}$. Here $g(x)\,\equiv\,a\overline{x}+bx\pmod{q}$, where $a,b,m$ and $k\ge 3$ are arbitrary integers, $(ab,q)=1$. The main result of the paper gives an asymptotic formula for the number of solutions for the case when $q$ is coprime …
Let $K_{q^n}(a)$ be a Kloosterman sum over the finite field $\F_{q^n}$ of characteristic $p$. In this note so called subfield conjecture is proved in case $p>3$: if $a\ne0$ belongs to the proper subfield $\F_q$ of $\F_{q^n}$, then $K_{q^n}(a)\ne-1$. This completes recent works on the subfield conjecture by Shparlinski, and Moisio and Lisonek. The problem is motivated by some applications to bent functions. Moreover, in the course of the proof a large class of translates of Dickson polynomials are shown to be irreducible.
Let $q$ be an integer $\geq 2$ and let $S_q(n)$ denote the sum of digits of $n$ in base $q$. For \[ α=[0;\overline{1,m}],\ m\geq 2, \] let $S_α(n)$ denote the sum of digits in the Ostrowski $α$-representation of $n$. Let $m_1,m_2\geq 2$ be integers with $$\gcd(q-1,m_1)=\gcd(m,m_2)=1.$$ We prove that there exists $δ>0$ such that for all integers $a_1,a_2$, \begin{eqnarray*} &&|\{0\leq n<N: S_{q}(n)\equiv a_1\pmod{m_1},\ S_α(n)\equiv a_2\pmod{m_2}\}| &=&\frac{N}{m_1m_2}+O(N^{1-δ}). \end{eqnarray*} The asymptotic relation implied by this equality was proved by Coquet, Rhin & Toffin and the equali…
Let $p$ be an odd prime and let $m\not\equiv 0\pmod p$ be a rational p-adic integer. In this paper we reveal the connection between quartic residues and the sum $\sum_{k=0}^{[p/4]}\binom{4k}{2k}\frac 1{m^k}$, where $[x]$ is the greatest integer not exceeding $x$. Let $q$ be a prime of the form $4k+1$ and so $q=a^2+b^2$ with $a,b\in\Bbb Z$. When $p\nmid ab(a^2-b^2)q$, we show that for $r=0,1,2,3$, $p^{\frac{q-1}4}\equiv (\frac ab)^r\pmod q$ if and only if $$\sum_{k=0}^{[p/4]}\binom{4k}{2k}\Big(\frac{a^2}{16q}\Big)^k\equiv (-1)^{\frac{p^2-1}8a+\frac{p-1}2\cdot \frac{q-1}4}\Big(\frac pq\Big) \Big…
We establish a new bound for the exponential sum \begin{eqnarray*} \sum_{x\in\mathcal{X}}\Big|\sum_{y\in \mathcal{Y}}γ(y)\exp(2πi a λ^{xy}/p)\Big|, \end{eqnarray*} where $λ$ is an element of the residue ring modulo a large prime number $p,$ $\mathcal{X}$ and $\mathcal{Y}$ are arbitrary subsets of the residue ring modulo $p-1$ and $γ(n)$ are any complex numbers with $|γ(n)| \le 1.$ In particular, we improve several previously known bounds.
Ramanujan sums have been studied and generalized by several authors. For example, Nowak studied these sums over quadratic number fields, and Grytczuk defined that on semigroups. In this note, we deduce some properties on sums of generalized Ramanujan sums and give examples on number fields. In particular, we have a relational expression between Ramanujan sums and residues of Dedekind zeta functions.
We obtain function field analogues of recent energy bounds on modular square roots and modular inversions and apply them to bounding some bilinear sums and to some questions regarding smooth and square-free polynomials in residue classes.
We obtain new results on additive properties of the set $$ I^{-1}= \{x^{-1}: \quad x\in I\} $$ where $I$ is an arbitrary interval in the field of residue classes modulo a large prime $p$. We combine our results with multilinear exponential sum estimates and obtain new results on incomplete multilinear Kloosterman sums.
Let $\F_p$ be the field of residue classes modulo a large prime $p$. The present paper is devoted to the problem of representability of elements of $\F_p$ as sums of fractions of the form $x/y$ with $x,y$ from short intervals of $\F_p$.
We bound double sums of Kloosterman sums over a finite field ${\mathbb F}_{q}$, with one or both parameters ranging over an affine space over its prime subfield ${\mathbb F}_p \subseteq {\mathbb F}_{q} $. These are finite fields analogues of a series of recent results by various authors in finite fields and residue rings. Our results are based on recent advances in additive combinatorics in arbitrary finite field.
Let $S(a,b)=12s(a,b)$, where $s(a,b)$ denotes the classical Dedekind sum. For a given denominator $q\in \mathbb N$, we study the numerators $k\in\mathbb Z$ of the values $k/q$, $(k,q)=1$, of Dedekind sums $S(a,b)$. Our main result says that if $k$ is such a numerator, then the whole residue class of $k$ modulo $(q^2-1)q$ consists of numerators of this kind. This fact reduces the task of finding all possible numerators $k$ to that of finding representatives for finitely many residue classes modulo $(q^2-1)q$. By means of the proof of this result we have determined all possible numerators $k$ fo…
Given an odd prime p, we prove a congruence on the p-residue of the Fermat quotient q_p(a) in base a with 0<a<p, which arises from a generalization of the Voronoi congruences and from some other congruences on sums and weighted sums of divided Bernoulli numbers. As an application in the base 2 case, we retrieve a congruence for the first generalized harmonic number of order (p-1)/2, a generalization originally due to Z-H. Sun of a classical congruence known since long for the harmonic number of order (p-1)/2 as a special case of the Lerch formula. We find a sharpening of the Voronoi congruence…
A set ${\cal A} \subseteq \Set{1,...,N}$ is of type $B_2$ if all sums $a+b$, with $a\ge b$, $a,b\in {\cal A}$, are distinct. It is well known that the largest such set is of size asymptotic to $N^{1/2}$. For a $B_2$ set ${\cal A}$ of this size we show that, under mild assumptions on the size of the modulus $m$ and on the difference $N^{1/2}-\Abs{\cal A}$ (these quantities should not be too large) the elements of ${\cal A}$ are uniformly distributed in the residue classes mod $m$. Quantitative estimates on how uniform the distribution is are also provided. This generalizes recent results of Lin…
For positive integers $p_1,p_2,\ldots,p_k,q$ with $q>1$, we define the Euler $T$-sum $T_{p_1p_2\cdots p_k,q}$ as the sum of those terms of the usual infinite series for the classical Euler sum $S_{p_1p_2\cdots p_k,q}$ with odd denominators. Like the Euler sums, the Euler $T$-sums can be evaluated according to the Contour integral and residue theorem. Using this fact, we obtain explicit formulas for Euler $T$-sums with repeated arguments analogous to those known for Euler sums. Euler $T$-sums can be written as rational linear combinations of the Hoffman $t$-values. Using known results for Hoffm…
In this paper we study products of quadratic residues modulo odd primes and prove some identities involving quadratic residues. For instance, let $p$ be an odd prime. We prove that if $p\equiv5\pmod8$, then $$\prod_{0<x<p/2,(\frac{x}{p})=1}x\equiv(-1)^{1+r}\pmod p,$$ where $(\frac{\cdot}{p})$ is the Legendre symbol and $r$ is the number of $4$-th power residues modulo $p$ in the interval $(0,p/2)$. Our work involves class number formula, quartic Gauss sums, Stickelberger's congruence and values of Dirichlet L-series at negative integers.
We study the variance of sums of the $k$-fold divisor function $d_k(n)$ over sparse arithmetic progressions, with averaging over both residue classes and moduli. In a restricted range, we confirm an averaged version of a recent conjecture about the asymptotics of this variance. This result is closely related to moments of Dirichlet $L$-functions and our proof relies on the asymptotic large sieve.
We consider a problem posed by Shparlinski, of giving nontrivial bounds for rational exponential sums over the arithmetic function $τ(n)$, counting the number of divisors of $n$. This is done using some ideas of Sathe concerning the distribution in residue classes of the function $ω(n)$, counting the number of prime factors of $n$, to bring the problem into a form where, for general modulus, we may apply a bound of Bourgain concerning exponential sums over subgroups of finite abelian groups and for prime modulus some results of Korobov and Shkredov.
Using a probabilistic model, based on random walks on the additive group $\mathbb{Z}/m\mathbb{Z}$, we prove that the values of certain real character sums are uniformly distributed in residue classes modulo $m$.
We prove, for a sufficiently small subset $\mathcal{A}$ of a prime residue field, an estimate on the number of solutions to the equation $(a_1-a_2)(a_3-a_4) = (a_5-a_6)(a_7-a_8)$ with all variables in $\mathcal{A}$. We then derive new bounds on trilinear exponential sums and on the total number of residues equaling the product of two differences of elements of $\mathcal{A}$. We 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.
Suppose that some harmonic analysis arguments have been invoked to show that the indicator function of a set of residue classes modulo some integer has a large Fourier coefficient. To get information about the structure of the set of residue classes, we then need a certain type of complementary result. A solution to this problem was given by Gregory Freiman in 1961, when he proved a lemma which relates the value of an exponential sum with the distribution of summands in semi-circles of the unit circle in the complex plane. Since then, Freiman's Lemma has been extended by several authors. Rathe…
Large sieve inequality for sums of Legendre symbols over short intervals
Université Jean Monnet, Centrale Lyon, INSA Lyon, Université Claude Bernard Lyon 1, CNRS, ICJ UMR5208（让·莫内大学、里昂中央理工学院、里昂国立应用科学学院、里昂第一大学、法国国家科学研究中心、ICJ UMR5208）；School of Mathematics and Statistics, University of New South Wales（新南威尔士大学数学与统计学院）；Department of Mathematics, University of Bristol（布里斯托大学数学系）；School of Mathematical Sciences, Shanghai Jiao Tong University（上海交通大学数学科学学院）
In this paper we give an essential treatment for estimating arbitrary integral power moments of Kloosterman sums over the residue class ring. For prime moduli we derive explicit estimates, and for prime-power moduli we prove concrete formulas using computations with Igusa zeta functions.
We obtain a new estimate for Kloosterman sum with primes $p\leqslant X$ to composite modulo $q$, that is, for the exponential sum of the type \[ \sum\limits_{p\leqslant X,\;p\,\nmid q}\exp{\biggl(\frac{2πi}{q}\bigl(a\overline{p}+bp\bigr)\,\biggr)},\quad (ab,q)=1,\quad p\overline{p}\equiv 1\pmod{q}, \] which is non-trivial in the case when $q^{\,3/4+\varepsilon}\leqslant X\ll q^{\,3/2}$. We also apply this estimate to the proof of solvability of some congruences with inverse prime residues $\pmod{q}$.
The rationality of the elliptic Gauss sum coefficient is shown. The following is a specific case of our argument. Let f(u)=sl((1-i)\varpi u), where sl() is the Gauss' lemniscatic sine and \varpi=2.62205... is the real period of the elliptic curve y^2=x^3-x, so that f(u) is an elliptic function relative to the period lattice Z[i]. Let πbe a primary prime of Z[i] such that norm(π)\equiv 13\mod 16. Let S be the quarter set mod πconsisting of quartic residues. Let us define G(π):=\sum_{ν\in S} f(ν/π) and \tildeπ:=\prod_{ν\in S} f(ν/π). The former G(π) is a typical example of elliptic Gauss sum; th…
We obtain a nontrivial bound on the number of solutions to the equation $$ A^{x_1} + \ldots + A^{x_ν} = A^{x_{ν+1}} + \ldots + A^{x_{2ν}}, \quad 1 \le x_1, \ldots,x_{2ν} \le τ, $$ with a fixed $n\times n$ matrix $A$ over a finite field $\mathbb F_q$ of $q$ elements of multiplicative order $τ$. We give applications of our result to obtaining a new bound of additive character sums with a matrix exponential function, which is nontrivial beyond the square-root threshold. For $n=2$ this equation has been considered by Kurlberg and Rudnick (2001) (for $ν=2$) and Bourgain (2005) (for large $ν$) in th…
We obtain several asymptotic formulas for the sum of the divisor function $τ(n)$ with $n \le x$ in an arithmetic progressions $n \equiv a \pmod q$ on average over $a$ from a set of several consecutive elements from set of reduced residues modulo $q$ and on average over arbitrary sets. The main goal is to obtain nontrivial results for $q \ge x^{2/3}$ with the small amount of averaging over $a$. We recall that for individual values of $a$ the limit of our current methods is $q \le x^{2/3-\varepsilon}$ for an arbitrary fixed $\varepsilon> 0$. Our method builds on an approach due to Blomer (2008) …
It has been conjectured by Sarkozy that with finitely many exceptions, the set of quadratic residues modulo a prime $p$ cannot be represented as a sumset $\{a+b\colon a\in A, b\in B\}$ with non-singleton sets $A,B\subset F_p$. The case $A=B$ of this conjecture has been recently established by Shkredov. The analogous problem for differences remains open: is it true that for all sufficiently large primes $p$, the set of quadratic residues modulo $p$ is not of the form $\{a'-a"\colon a',a"\in A,\,a'\ne a"\}$ with $A\subset F_p$? We attack here a presumably more tractable variant of this problem, …
In this paper, we first find the distribution of nth power residues modulo a prime $p$ by analyzing sums involving Dirichlet characters. We then extend this method to characterize the distribution of powers in arbitrary finite fields.
We use character sums to confirm several recent conjectures of V. I. Arnold on the uniformity of distribution properties of a certain dynamical system in a finite field. On the other hand, we show that some conjectures are wrong. We also analyze several other conjectures of V. I. Arnold related to the orbit length of similar dynamical systems in residue rings and outline possible ways to prove them. We also show that some of them require further tuning.
In this paper we establish connections between covers of $\mathbb Z$ by residue classes and subset sums in a field. Suppose that $A_0=\{a_s(n_s)\}_{s=0}^k$ covers each integer at least $p$ times with the residue class $a_0(n_0)=a_0+n_0\mathbb Z$ irredundant, where $p$ is a prime not dividing any of $n_1,\ldots,n_k$. Let $m_1,\ldots,m_k\in\mathbb Z$ be relatively prime to $n_1,\ldots,n_k$ respectively. For any $c,c_1,\ldots,c_k\in\mathbb Z/p\mathbb Z$ with $c_1\cdots c_k\not=0$, we show that the set $$\bigg\{\bigg\{\sum_{s\in I}\frac{m_s}{n_s}\bigg\}:\, I\subseteq\{1,\ldots,k\} \ \mbox{and}\ \s…
A. Booker and C. Pomerance (2017) have shown that any residue class modulo a prime $p\ge 11$ can be represented by a positive $p$-smooth square-free integer $s = p^{O(\log p)}$ with all prime factors up to $p$ and conjectured that in fact one can find such $s$ with $s = p^{O(1)}$. Using bounds on double Kloosterman sums due to M. Z. Garaev (2010) we prove this conjecture in a stronger form $s \le p^{3/2 + o(1)}$ and also consider more general versions of this question replacing $p$-smoothness of $s$ by the stronger condition of $p^α$-smoothness. Using bounds on multiplicative character sums an…
