Measure rigidity is a branch of ergodic theory that has recently contributed to the solution of some fundamental problems in number theory and mathematical physics. Examples are proofs of quantitative versions of the Oppenheim conjecture, related questions on the spacings between the values of quadratic forms, a proof of quantum unique ergodicity for certain classes of hyperbolic surfaces, and an approach to the Littlewood conjecture on the nonexistence of multiplicatively badly approximable numbers.
In these lectures we discuss a few simple applications of one of the central results in measure rigidity: Ratner's theorem. We shall investigate the statistical properties of certain number theoretic sequences, specifically the fractional parts of $mα$, $m=1,2,3,...$, (a classical, well understood problem) and of $\sqrt{mα}$ (as recently studied by Elkies and McMullen). By exploiting equidistribution results on a certain homogeneous space, we will show that the statistical properties of these sequences can exhibit significant deviations from those of independent random variables. The ``randomness'' of other, more generic sequences such as $m^2α$ and $2^m α$ mod 1 has been studied extensively.
These notes are based on lectures presented at the Institute Henri Poincare Paris, June 2005, and at the summer school `Equidistribution in number theory', CRM Montreal, July 2005. The author gratefully acknowledges support by an EPSRC Advanced Research Fellowship.
Let G=ASL(2,R) be the affine special linear group of the plane, and set Gamma=ASL(2,Z). Building on recent work of Strömbergsson we prove a rate of equidistribution for the orbits of a certain 1-dimensional unipotent flow of Gamma\G, which projects to a closed horocycle in the unit tangent bundle to the modular surface. We use this to answer a question of Elkies and McMullen by making effective the convergence of the gap distribution of \sqrt{n} modulo 1.
Unipotent flows are well-behaved dynamical systems. In particular, Marina Ratner has shown that the closure of every orbit for such a flow is of a nice algebraic (or geometric) form. After presenting some consequences of this important theorem, these lectures explain the main ideas of the proof. Some algebraic technicalities will be pushed to the background. Chapter 1 is the main part of the book. It is intended for a fairly general audience, and provides an elementary introduction to the subject, by presenting examples that illustrate the theorem, some of its applications, and the main ideas …
We study multiplicative dependence between elements in orbits ofalgebraic dynamical systems over number fields modulo a finitely generated multiplicative subgroup of the field. We obtain a series of results, many of which may be viewed as a blend of Northcott's theorem on boundedness of preperiodic points and Siegel's theorem on finiteness of solutions to $S$-unit equations.
We give a simplified and a direct proof of a special case of Ratner's theorem on closures and uniform distribution of individual orbits of unipotent flows; namely, the case of orbits of the diagonally embedded unipotent subgroup acting on $SL(2,K)/Γ_1\times ...\times SL(2,K)/Γ_n$, where $K$ is a locally compact field of characteristic 0 and each $Γ_i$ is a cocompact discrete subgroup of $SL(2,K)$. This special case of Ratner's theorem plays a crucial role in the proofs of uniform distribution of Heegner points by Vatsal, and Mazur conjecture on Heegner points by C. Cornut; and their generaliza…
Let $p>3$ be a prime, and let $(\frac{\cdot}p)$ be the Legendre symbol. Let $b\in\mathbb Z$ and $\varepsilon\in\{\pm 1\}$. We mainly prove that $$\left|\left\{N_p(a,b):\ 1<a<p\ \text{and}\ \left(\frac ap\right)=\varepsilon\right\}\right|=\frac{3-(\frac{-1}p)}2,$$ where $N_p(a,b)$ is the number of positive integers $x<p/2$ with $\{x^2+b\}_p>\{ax^2+b\}_p$, and $\{m\}_p$ with $m\in\mathbb{Z}$ is the least nonnegative residue of $m$ modulo $p$.
For a number field $K$ with ring of integers ${\mathcal O}_K$, we prove an analogue over finite rings of the form ${\mathcal O}_K/{\mathcal P}^m$ of the Fundamental Theorem on the Fourier transform of a relative invariant of prehomogeneous vector spaces, where ${\mathcal P}$ is a big enough prime ideal of ${\mathcal O}_K$ and $m>1$. In the appendix, F. Sato gives an application of the Theorems A, B and the Theorems A, B, C in J. Denef and A. Gyoja [\emph{Character sums associated to prehomogeneous vector spaces}, Compos. Math., \textbf{113} (1998) 237--346] to the functional equation of $L$-fu…
We consider the distribution of spacings between consecutive elements in subsets of Z/qZ where q is highly composite and the subsets are defined via the Chinese remainder theorem. We give a sufficient criterion for the spacing distribution to be Poissonian as the number of prime factors of q tends to infinity, and as an application we show that the value set of a generic polynomial modulo q have Poisson spacings. We also study the spacings of subsets of Z/q_1q_2Z that are created via the Chinese remainder theorem from subsets of Z/q_1Z and Z/q_2Z (for q_1,q_2 coprime), and give criteria for wh…
The reduction modulo $p$ of a family of lacunary integer polynomials, associated with the dynamical zeta function $ζ_β(z)$ of the $β$-shift, for $β> 1$ close to one, is investigated. We briefly recall how this family is correlated to the problem of Lehmer. A variety of questions is raised about their numbers of zeroes in $\mathbb{F}_p$ and their factorizations, via Kronecker's Average Value Theorem (viewed as an analog of classical Theorems of Uniform Distribution Theory). These questions are partially answered using results of Schinzel, revisited by Sawin, Shusterman and Stoll, and density th…
We give results characterising ternary Kloosterman sums modulo 9 and 27. This leads to a complete characterisation of values that ternary Kloosterman sums assume modulo 18 and 54. The proofs uses Stickelberger's theorem, the Gross-Koblitz formula and Fourier analysis.
$ $The aim of this thesis is to lower the bound on square-free primitive roots modulo primes. Let $g^{\Box}(p)$ be the least square-free primitive root modulo $p$. We have proven the following two theorems. Theorem 0.1. $$g^{\Box}(p) < p^{0.88}\quad\text{for all primes }p.$$ Theorem 0.2. $$g^{\Box}(p) < p^{0.63093}\quad\text{for all primes } p < 2.5\times10^{15} \text{ and }p > 9.63\times10^{65}.$$ Theorem 0.1 shows an improvement in the best known bound while Theorem 0.2 shows for which primes we can prove the theoretical lower bound. After some introductory information in Chapter 1, we will …
In this note we prove that {equation*} {np^s\choose mp^s+r}\equiv (-1)^{r-1}r^{-1}(m+1){n\choose m+1}p^s \pmod{p^{s+1}} {equation*} where $p$ is any prime, $n$, $m$, $s$ and $r$ are nonnegative integers such that $n\ge m$, $s\ge 1$, $1\le r\le p^s-1$ and $r$ is not divisible by $p$. We derive a proof by induction using a multiple application of Lucas' theorem and two basic binomial coefficient identities. As an application, we prove that a similar congruence for a prime $p\ge 5$ established in 1992 by D. F. Bailey holds for each prime $p$.
Let $p$ be a prime, and let $k,n,m,n_0$ and $m_0$ be nonnegative integers such that $k\ge 1$, and $_0$ and $m_0$ are both less than $p$. K. Davis and W. Webb established that for a prime $p\ge 5$ the following variation of Lucas' Theorem modulo prime powers holds $$ {np^k +n_0 \choose mp^k+m_0}\equiv{np^{\lfloor(k-1)/3\rfloor} \choose mp^{\lfloor(k-1)/3\rfloor}} {n_0 \choose m_0} \pmod{p^k}. $$ In the proof the authors used their earlier result that present a generalized version of Lucas' Theorem. In this paper we present a a simple inductive proof of the above congruence. Our proof is based o…
In this article, we follow Hida's approach to establish an analogue of Washington's theorem on the non-vanishing modulo p of Hecke L-values for CM fields with anticyclotomic twists.
First, we present a new proof of Glaisher's formula dating from 1900 and concerning Wilson's theorem modulo p^2. Our proof uses p-adic numbers and Faulhaber's formula for the sums of powers (17th century), as well as more recent results on Faulhaber's coefficients obtained by Gessel and Viennot. Second, by using our method, we find a simpler proof than Sun's proof regarding a formula for (p-1)! modulo p^3, and one that can be generalized to higher powers of p. Third, we can derive from our method a way to compute the Stirling numbers modulo p^3, thus improving Glaisher and Sun's own results fr…
We discuss a clean level lowering theorem modulo prime powers for weight $2$ cusp forms. Furthermore, we illustrate how this can be used to completely solve certain twisted Fermat equations $ax^n+by^n+cz^n=0$.
Freiman's 2.4-Theorem states that any set $A \subset \mathbb{Z}_p$ satisfying $|2A| \leq 2.4|A| - 3 $ and $|A| < p/35$ can be covered by an arithmetic progression of length at most $|2A| - |A| + 1$. A more general result of Green and Ruzsa implies that this covering property holds for any set satisfying $|2A| \leq 3|A| - 4$ as long as the rather strong density requirement $|A| < p/10^{215}$ is satisfied. We present a version of this statement that allows for sets satisfying $|2A| \leq 2.48|A| - 7$ with the more modest density requirement of $|A| < p/10^{10}$.
This paper studies so-called "null polynomials modulo m", i.e., polynomials with integer coefficients that satisfy f(x)=0 (mod m) for any integer x. The study on null polynomials is helpful to reduce congruences of higher degrees modulo m and to enumerate equivalent polynomial functions modulo m, i.e., functions over Z_m={0, ..., m-1} generated by integer polynomials. The most well-known null polynomial is f(x)=x^p-x modulo a prime p. After pointing out that null polynomials modulo a composite can be studied by handling null polynomials modulo each prime power, this paper mainly focuses on nul…
For positive integers $q$, Dirichlet's theorem states that there are infinitely many primes in each reduced residue class modulo $q$. A stronger form of the theorem states that the primes are equidistributed among the $φ(q)$ reduced residue classes modulo $q$. This paper considers patterns of sequences of consecutive primes $(p_n, p_{n+1}, \ldots, p_{n+k})$ modulo $q$. Numerical evidence suggests a preference for certain prime patterns. For example, computed frequencies of the pattern $(a,a)$ modulo $q$ up to $x$ are much less than the expected frequency $π(x)/φ(q)^2$. We begin to rigorously c…
In 1947 Fine obtained an expression for the number of binomial coefficients on row n of Pascal's triangle that are nonzero modulo p. In this paper we use Kummer's theorem to generalize Fine's theorem to prime powers, expressing the number of nonzero binomial coefficients modulo p^alpha as a sum over certain integer partitions. For fixed alpha, this expression can be rewritten to show explicit dependence on the number of occurrences of each subword in the base-p representation of n.
Let $\mathcal{K}(a)$ denote the Kloosterman sum on the finite field of order $2^n$. We give a simple characterization of $\mathcal{K}(a)$ modulo 16, in terms of the trace of $a$ and one other function. We also give a characterization of $\mathcal{K}(a)$ modulo 64 in terms of a different function, which we call the lifted trace. Our proofs use Fourier analysis, Stickelberger's theorem and the Gross-Koblitz formula.
Building on work of Zhi-Hong Sun, we establish congruences for the special harmonic numbers $H_\lfloor p/9 \rfloor$ and $H_{\lfloor p/18 \rfloor}$ modulo $p$, which contain respectively three and four distinct arithmetic components. We also obtain a complete determination modulo $p$ of the corresponding families of sums of reciprocals of the type studied by Dilcher and Skula. Applications to the first case of Fermat's Last Theorem are considered.
For a nonnegative integer $n$, and a prime $\wp$ in $\mathbb{F}_q[T]$, we prove a result that provides a method for computing the number of integers $m$ with $0 \le m \le n$ for which the Carlitz binomial coefficients $\binom{n}{m}_C$ fall into each of the residue classes modulo $\wp$. Our main result can be viewed as a function field analogue of the Garfield-Wilf theorem.
In this paper we study the distribution of very short sequences of inversive congruential pseudorandom numbers modulo $2^t$. We derive a new bound on exponential sums with such sequences and use it to give estimate their discrepancy. The technique we use, based the method of N. M. Korobov (1972) of estimating double Weyl sums and a fully explicit form of the Vinogradov mean value theorem due to K. Ford (2002), has never been used in this area and is very likely to find further applications.
Recently, Bruinier and Ono classified cusp forms $f(z) := \sum_{n=0}^{\infty} a_f(n)q ^n \in S_{λ+1/2}(Γ_0(N),χ)\cap \mathbb{Z}[[q]]$ that does not satisfy a certain distribution property for modulo odd primes $p$. In this paper, using Rankin-Cohen Bracket, we extend this result to modular forms of half integral weight for primes $p \geq 5$. As applications of our main theorem we derive distribution properties, for modulo primes $p\geq5$, of traces of singular moduli and Hurwitz class number. We also study an analogue of Newman's conjecture for overpartitions.
Given an integer $q$ and a polynomial $f\in \mathbb Z_{q}[X]$ of degree $d$ with coefficients in the residue ring $\mathbb Z_q=\mathbb Z/q\mathbb Z,$ we obtain new results concerning the number of solutions to congruences of the form $$y\equiv f(x) \pmod{q},$$ with integer variables lying in some cube $\mathcal B$ of side length $H$. Our argument uses ideas of Cilleruelo, Garaev, Ostafe and Shparlinski which reduces the problem to the Vinogradov mean value theorem and a lattice point counting problem. We treat the lattice point problem differently using transference principles from the Geometr…
In this work we provide a level raising theorem for $\mod λ^n$ modular Galois representations. It allows one to see such a Galois representation that is modular of level $N$, weight 2 and trivial Nebentypus as one that is modular of level $Np$, for a prime $p$ coprime to $N$, when a certain local condition at $p$ is satisfied. It is a generalization of a result of Ribet concerning $\mod \ell$ Galois representations.
In the paper of Tor Helleseth and Victor Zinoviev (Designs, Codes and Cryptography, \textbf{17}, 269-288(1999)), the number of solutions of the system of equations from $ Z_{4} $-linear Goethals codes $ G_{4} $ was determined and stated in Theorem 4. We found that Theorem 4 is wrong for $ m $ even. In this note, we complete Theorem 4, and give new divisibility modulo 12 of the Kloosterman sums deduced from Theorem 4, which is different from the result of the same authors for $ K(a^{4}+a^{3}) $ modulo 12.
Let $f(x)$ be an irreducible polynomial with integer coefficients of degree at least two. Hooley proved that the roots of the congruence equation $f(x)\equiv 0\mod n$ is uniformly distributed. as a parallel of Hooley's theorem under ideal theoretical setting, we prove the uniformity of the distribution of residues of an algebraic number modulo degree one ideals. Then using this result we show that the roots of a system of polynomial congruences are uniformly distributed. Finally, the distribution of digits of n-adic expansions of an algebraic number is discussed.
We study the distribution modulo $1$ of the values taken on the integers of $r$ linear forms in $d$ variables with random coefficients. We obtain quenched and annealed central limit theorems for the number of simultaneous hits into shrinking targets of radii $n^{-\frac{r}{d}}$. By the Khintchine-Groshev theorem on Diophantine approximations, $\frac{r}{d}$ is the critical exponent for the infinite number of hits.
Distribution modulo one of linear recurrent sequences
East China Normal University（华东师范大学）；Morningside Center of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院晨兴数学中心）；University of Chinese Academy of Sciences（中国科学院大学）
Let ${\mathbb Z}_p$ denote the ring of all $p$-adic integers and call $${\mathcal U}=\{(x_1,\ldots,x_n):\,a_1x_1+\ldots+a_nx_n+b=0\}$$ a hyperplane over ${\mathbb Z}_p^n$, where at least one of $a_1,\ldots,a_n$ is not divisible by $p$. We prove that if a sufficiently regular $n$-variable function is zero modulo $p^r$ over some suitable collection of $r$ hyperplanes, then it is zero modulo $p^r$ over the whole ${\mathbb Z}_p^n$. We provide various applications of this general criterion by establishing several $p$-adic analogues of hypergeometric identities.
A well-known generalisation of positional numeration systems is the case where the base is the residue class of $x$ modulo a given polynomial $f(x)$ with coefficients in (for example) the integers, and where we try to construct finite expansions for all residue classes modulo $f(x)$, using a suitably chosen digit set. We give precise conditions under which direct or fibred products of two such polynomial number systems are again of the same form. The main tool is a general form of the Chinese Remainder Theorem. We give applications to simultaneous number systems in the integers.
We present new generalizations of Olson's theorem and of a consequence of Alon's Combinatorial Nullstellensatz. These enable us to extend some of their combinatorial applications with conditions modulo primes to conditions modulo prime powers. We analyze computational search problems corresponding to these kinds of combinatorial questions and we prove that the problem of finding degree-constrained subgraphs modulo $2^d$ such as $2^d$-divisible subgraphs and the search problem corresponding to the Combinatorial Nullstellensatz over $\mathbb{F}_2$ belong to the complexity class Polynomial Parity…
In terms of the creative microscoping method recently introduced by Guo and Zudilin and the Chinese remainder theorem for coprime polynomials, we establish a $q$-supercongruence with two parameters modulo $[n]Φ_n(q)^3$. Here $[n]=(1-q^n)/(1-q)$ and $Φ_n(q)$ is the $n$-th cyclotomic polynomial in $q$. In particular, we confirm a recent conjecture of Guo and give a complete $q$-analogue of Long's supercongruence. The latter is also a generalization of a recent $q$-supercongruence obtained by Guo and Schlosser.
The system of equations \[ u_1p_1^2 + \ldots + u_sp_s^2 = 0 \] \[ v_1p_1^3 + \ldots + v_sp_s^3 = 0 \] has prime solutions $(p_1, \ldots, p_s)$ for $s \geq 12$, assuming that the system has solutions modulo each prime $p$. This is proved via the Hardy-Littlewood circle method, building on Wooley's work on the corresponding system over the integers and recent results on Vinogradov's mean value theorem. Additionally, a set of sufficient conditions for local solvability is given: If both equations are solvable modulo 2, the quadratic equation is solvable modulo 3, and for each prime $p$ at least 7…
Koksma's equidistribution theorem from 1935 states that for Lebesgue almost every $α>1$, the fractional parts of the geometric progression $(α^{n})_{n\geq1}$ are equidistributed modulo one. In the present paper we sharpen this result by showing that for almost every $α>1$, the correlations of all finite orders and hence the normalized gaps of $(α^{n})_{n\geq1}$ mod 1 have a Poissonian limit distribution, thereby resolving a conjecture of the two first named authors. While an earlier approach used probabilistic methods in the form of martingale approximation, our reasoning in the present paper …
We give q-analogues of Wilson's theorem for the primes congruent 1 and 3 modulo 4 respectively. And q-analogues of two congruences due to Mordell and Chowla are also established.
A classical result of A. Fleck states that if p is a prime, and n>0 and r are integers, then $$\sum_{k=r(mod p)}\binom {n}{k}(-1)^k=0 (mod p^{[(n-1)/(p-1)]}).$$ Recently R. M. Wilson used Fleck's congruence and Weisman's extension to present a useful lemma on polynomials modulo prime powers, and applied this lemma to reprove the Ax-Katz theorem on solutions of congruences modulo p and deduce various results on codewords in p-ary linear codes with weights. In light of the recent generalizations of Fleck's congruence given by D. Wan, and D. M. Davis and Z. W. Sun, we obtain new extensions of Wil…
For $k\geq i\geq 1$, let $B_{k,i}(n)$ denote the number of partitions of $n$ such that part 1 appears at most $i-1$ times, two consecutive integers l and $l+1$ appear at most $k-1$ times and if l and $l+1$ appear exactly $k-1$ times then the total sum of the parts l and $l+1$ is congruent to $i-1$ modulo 2. Let $A_{k,i}(n)$ denote the number of partitions with parts not congruent to $i$, $2k-i$ and $2k$ modulo $2k$. Bressoud's theorem states that $A_{k,i}(n)=B_{k,i}(n)$. Corteel, Lovejoy, and Mallet found an overpartition analogue of Bressoud's theorem for $i=1$, that is, for partitions not co…
As society becomes more reliant on computers, cryptographic security becomes increasingly important. Current encryption schemes include the ElGamal signature scheme, which depends on the complexity of the discrete logarithm problem. It is thought that the functions that such schemes use have inverses that are computationally intractable. In relation to this, we are interested in counting the solutions to a generalization of the discrete logarithm problem modulo a prime power. This is achieved by interpolating to p-adic functions, and using Hensel's lemma, or other methods in the case of singul…
Given a matrix $A\in \mathrm{GL}_d(\mathbb{Z})$. We study the pseudorandomness of vectors $\mathbf{u}_n$ generated by a linear recurrent relation of the form $$ \mathbf{u}_{n+1} \equiv A \mathbf{u}_n \pmod {p^t}, \qquad n = 0, 1, \ldots, $$ modulo $p^t$ with a fixed prime $p$ and sufficiently large integer $t \geq 1$. We study such sequences over very short segments of length which is not accessible via previously used methods. Our technique is based on the method of N. M. Korobov (1972) of estimating double Weyl sums and a fully explicit form of the Vinogradov mean value theorem due to K. For…
Let k be a field not of characteristic two and L be a set of almost all rational primes invertible in k. Suppose we have a variety X/k and strictly compatible system {M_ell -> X : ell in L} of constructible F_ell-sheaves. If the system is orthogonally or symplectically self-dual, then the geometric monodromy group of M_ell is a subgroup of a corresponding isometry group G_ell over F_ell, and we say it has big monodromy if it contains the derived subgroup DG_ell=[G_ell,G_ell]. We prove a theorem which gives sufficient conditions for M_ell to have big monodromy. We apply the theorem to explicit …
In this paper we study $b_5(n)$, the $5$-regular partitions of $n$. Using the theory of modular forms, we prove several theorems on the divisibility and distribution properties of $b_5(n)$ modulo prime $m\geq5$. In particular, we prove that there are infinitely many Ramanujan-type congruences modulo prime $m\geq5$.
The equation $x^2 + 1 = 0\mod p$ has solutions whenever $p = 2$ or $4n + 1$. A famous theorem of Fermat says that these primes are exactly the ones that can be described as a sum of two squares. That the roots of the former equation are equidistributed is a beautiful theorem of Duke, Friedlander and Iwaniec from 1995. We show that a subsequence of the roots of the equation remains equidistributed even when one adds a restriction on the primes which has to do with the angle in the plane formed by their corresponding representation as a sum of squares. Similar to Duke, Friedlander and Iwaniec, w…
For a fixed number field $K$, we consider the mean square error in estimating the number of primes with norm congruent to $a$ modulo $q$ by the Chebotarëv Density Theorem when averaging over all $q\le Q$ and all appropriate $a$. Using a large sieve inequality, we obtain an upper bound similar to the Barban-Davenport-Halberstam Theorem.
In this paper, we mainly give a general explicit form of Cassels' $p$-adic embedding theorem for number fields. We also give its refined form in the case of cyclotomic fields. As a byproduct, given an irreducible polynomial $f$ over $Z$, we give a general unconditional upper bound for the smallest prime number $p$ such that $f$ has a simple root modulo $p$.
Let $\mathbb{A}[n]$ be the group of $n$-torsion points of a commutative algebraic group $\mathbb{A}$ defined over a number field $F$. For a prime ideal $\mathfrak{p}$, we let $N_{\mathfrak{p}}(\mathbb{A}[n])$ be the number of $\mathbb{F}_\mathfrak{p}$-solutions of the system of polynomial equations defining $\mathbb{A}[n]$ when reduced modulo $\mathfrak{p}$. Here, $\mathbb{F}_{\mathfrak{p}}$ is the residue field at $\mathfrak{p}$. Let $π_F(x)$ denote the number of primes $\mathfrak{p}$ of $F$ whose norm $N(\mathfrak{p})$ do not exceed $x$. We then, for algebraic groups of dimension one, comput…
We define a new congruence relation on the set of integers, leading to a group similar to the multiplicative group of integers modulo $n$. It makes use of a symmetry almost omnipresent in modular multiplications and halves the number of residue classes. Using it, we are able to give an elegant description of some results due to Carl Schick, others are reduced to well-known theorems from algebra and number theory. Many concepts from number theory such as quadratic residues and primitive roots are equally applicable. It brings noticeable advantages in studying powers of odd primes, and in partic…
