For a positive integer $N$, we define the N-rank of a non singular integer $d\times d$ matrix $A$ to be the maximum integer $r$ such that there exists a minor of order $r$ whose determinant is not divisible by $N$. Given a positive integer $r$, we study the growth of the minumum integer $k$, such that $A^k-I$ has N-rank at most $r$, as a function of $N$. We show that this integer $k$ goes to infinity faster than $\log N$ if and only if for every eigenvalue $λ$ which is not a root of unity, the sum of the dimensions of the eigenspaces relative to eigenvalues which are multiplicatively dependent with $λ$ and are not roots of unity, plus the dimensions of the eigenspaces relative to eigenvalues which are roots of unity, does not exceed $d-r-1$. This result will be applied to recover a recent theorem of Luca and Shparlinski which states that the group of rational points of an ordinary elliptic curve $E$ over a finite field with $q^n$ elements is almost cyclic, in a sense to be defined, when $n$ goes to infinity. We will also extend this result to the product of two elliptic curves over a finite field and show that the orders of the groups of $\mathbb{F}_{q^n}-$rational points of two non isogenous elliptic curves are almost coprime when $n$ approaches infinity.
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…
We proved recently (see \cite{lhgarasu}) the result on the title for odd prime divisors of such an $n.$ The result implies for many $n's$, more precisely, for an infinity of $n$'s with an arbitrary fixed number of prime divisors, the inexistence of circulant Hadamard matrices, and the inexistence of Barker sequences of length $n >13$. The proof used a result of Arasu. It turns out that there is another, shorter proof, of the more general result that includes the prime $p=2.$ This new proof is based on a result of Brock (see \cite[Theorem 3.1]{brock}), and besides that, requires just the defini…
The quantum cat map is a model for a quantum system with underlying chaotic dynamics. In this paper we study the matrix elements of smooth observables in this model, when taking arithmetic symmetries into account. We give explicit formulas for the matrix elements as certain exponential sums. With these formulas we can show that there are sequences of eigenfunctions for which the matrix elements decay significantly slower then was previously conjectured. We also prove a limiting distribution for the fluctuation of the normalized matrix elements around their average.
We prove the modulo $p$ and modulo $p^2$ cases of Igusa's conjecture on exponential sums. This conjecture predicts specific uniform bounds in the homogeneous polynomial case of exponential sums modulo $p^m$ when $p$ and $m$ vary. We introduce the motivic oscillation index of a polynomial $f$ and prove the stronger, analogue bounds for $m=1,2$ using this index instead of the original bounds. The modulo $p^2$ case of our bounds holds for all polynomials; the modulo $p$ case holds for homogeneous polynomials and under extra conditions also for nonhomogeneous polynomials. We obtain natural lower b…
We present lower bounds for the orbit length of reduction modulo primes of parametric polynomial dynamical systems defined over the integers, under a suitable hypothesis on its set of preperiodic points over $\mathbb C$. Applying recent results of Baker and DeMarco~(2011) and of Ghioca, Krieger, Nguyen and Ye~(2017), we obtain explicit families of parametric polynomials and initial points such that the reductions modulo primes have long orbits, for all but a finite number of values of the parameters. This generalizes a previous lower bound due to Chang~(2015). As a by-product, we also slighly …
Given a prime $p$ and a positive integer $k$, let $\mathrm{M}_{n}(\mathbb{Z}/p^{k}\mathbb{Z})$ be the ring of $n \times n$ matrices over $\mathbb{Z}/p^{k}\mathbb{Z}$. We consider the number of solutions $X \in \mathrm{M}_{n}(\mathbb{Z}/p^{k}\mathbb{Z})$ to the polynomial equation $P(X) = 0$, where $P(t)$ is a monic polynomial in $(\mathbb{Z}/p^{k}\mathbb{Z})[t]$ whose reduction modulo $p$ is square-free over the finite field $\mathbb{F}_{p}$ of $p$ elements. Noting that $P(X) = 0$ if and only if $\mathrm{cok}(P(X)) \simeq (\mathbb{Z}/p^{k}\mathbb{Z})^{n}$, we give a conjectural generalization …
Assuming the Generalized Riemann Hypothesis, we prove the following: If b is an integer greater than one, then the multiplicative order of b modulo N is larger than N^(1-ε) for all N in a density one subset of the integers. If A is a hyperbolic unimodular matrix with integer coefficients, then the order of A modulo p is greater than p^(1-ε) for all p in a density one subset of the primes. Moreover, the order of A modulo N is greater than N^(1-ε) for all N in a density one subset of the integers.
$ $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 …
Let $A$ be a set in an abelian group $G$. For integers $h,r \geq 1$ the generalized $h$-fold sumset, denoted by $h^{(r)}A$, is the set of sums of $h$ elements of $A$, where each element appears in the sum at most $r$ times. If $G=\mathbb{Z}$ lower bounds for $|h^{(r)}A|$ are known, as well as the structure of the sets of integers for which $|h^{(r)}A|$ is minimal. In this paper we generalize this result by giving a lower bound for $|h^{(r)}A|$ when $G=\mathbb{Z}/p\mathbb{Z}$ for a prime $p$, and show new proofs for the direct and inverse problems in $\mathbb{Z}$.
We describe a class of matrices whose determinants are trivial to compute. A nice example of such a matrix is given by considering the symmetric matrix with entries {i+j choose i} (mod 2) in {0,1}, 0 <= i,j < n the binomial coefficients modulo 2 represented by {0,1}. The determinant of the above matrix is then closely related to the Thue-Morse sequence counting digits modulo 2 of binary integers.
We obtain new lower bounds on the number of smooth squarefree integers up to $x$ in residue classes modulo a prime $p$, relatively large compared to $x$, which in some ranges of $p$ and $x$ improve that of A. Balog and C. Pomerance (1992). We also estimate the smallest squarefull number in almost all residue classes modulo a prime $p$.
For a nonsingular integer matrix A, we study the growth of the order of A modulo N. We say that a matrix is exceptional if it is diagonalizable, and a power of the matrix has all eigenvalues equal to powers of a single rational integer, or all eigenvalues are powers of a single unit in a real quadratic field. For exceptional matrices, it is easily seen that there are arbitrarily large values of N for which the order of A modulo N is logarithmically small. In contrast, we show that if the matrix is not exceptional, then the order of A modulo N goes to infinity faster than any constant multiple …
We consider the periods of the linear congruential and the power generators modulo $n$ and, for fixed choices of initial parameters, give lower bounds that hold for ``most'' $n$ when $n$ ranges over three different sets: the set of primes, the set of products of two primes (of similar size), and the set of all integers. For most $n$ in these sets, the period is at least $n^{1/2+ε(n)}$ for any monotone function $ε(n)$ tending to zero as $n$ tends to infinity. Assuming the Generalized Riemann Hypothesis, for most $n$ in these sets the period is greater than $n^{1-ε}$ for any $ε>0$. Moreover, the…
Let K/Q be Galois, and let eta in K* whose conjugates are multiplicatively independent. For a prime p, unramified, prime to eta, let np be the residue degree of p and gp the number of P I p, then let o\_P(eta) and o\_p(eta) be the orders of eta modulo P and p, respectively.Using Frobenius automorphisms, we show that for all p\textgreater{}\textgreater{}0, some explicit divisors of p^(np)-1 cannot realize o\_P(eta) nor o\_p(eta), and we give a lower bound of o\_p(eta).Then we obtain that, for all p\textgreater{}\textgreater{}0 such that np \textgreater{}1, Prob(o\_p(eta)\textless{}p) $\le$ 1/p^…
We solve a problem due to Recamán about the lower bound behavior of the maximum possible length among all arithmetic progressions in the least reduced residue system modulo $n$, as $n \to \infty$.
Euler's totient function, $φ(n)$, which counts how many of $0,1,\dots,n-1$ are coprime to $n$, has an explicit asymptotic lower bound of $n/\log \log n$, modulo some constant. In this note, we generalise $φ$; given an irreducible integer polynomial $P$, we define the arithmetic function $φ_P(n)$ that counts the amount of numbers among $P(0),P(1),\dots,P(n-1)$ that are coprime to $n$. We also provide an asymptotic lower bound for $φ_P(n)$.
We establish an analogue of the classical Polya-Vinogradov inequality for $GL(2, \F_p)$, where $p$ is a prime. In the process, we compute the `singular' Gauss sums for $GL(2, \F_p)$. As an application, we show that the collection of elements in $GL(2,\Z)$ whose reduction modulo $p$ are of maximal order in $GL(2, \F_p)$ and whose matrix entries are bounded by $x$ has the expected size as soon as $x\gg p^{1/2+\ep}$ for any $\ep>0$. In particular, there exist elements in $GL(2,\Z)$ with matrix entries that are of the order $O(p^{1/2+\ep})$ whose reduction modulo $p$ are primitive elements.
Much work has been done attempting to understand the dynamic behaviour of the so-called "3x+1" function. It is known that finite sequences of iterations with a given length and a given number of odd terms have some combinatorial properties modulo powers of two. In this paper, we formulate a new hypothesis asserting that the first terms of those sequences have a lower bound which depends on the binary entropy of the "ones-ratio". It is in agreement with all computations so far. Furthermore it implies accurate upper bounds for the total stopping time and the maximum excursion of an integer. Thes…
An important result of H. Weyl states that for every sequence $\left(a_{n}\right)_{n \geq 1}$ of distinct positive integers the sequence of fractional parts of $\left(a_{n} α\right)_{n\geq 1}$ is uniformly distributed modulo one for almost all $α$. However, in general it is a very hard problem to calculate the precise order of convergence of the discrepancy of $\left(\left\{a_{n} α\right\}\right)_{n \geq 1}$ for almost all $α$. In particular it is very difficult to give sharp lower bounds for the speed of convergence. Until now this was only carried out for lacunary sequences $\left(a_{n}\righ…
Solving Modular Linear Systems with a Constraint by parallel decomposition of the Smith form and extended Euclidean division modulo powers of primes divisors
