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 theorems (Frobenius, Chebotarev, Serre, Rosen). These questions arise from the search for the existence of integer polynomials of Mahler measure > 1 less than the smallest Salem number 1.176280. Explicit connection with modular forms (or modular representations) of the numbers of zeroes of these polynomials in $\mathbb{F}_p$ is obtained in a few cases. In general it is expected since it must exist according to the Langlands program.
An integral power series is called lacunary modulo $M$ if almost all of its coefficients are divisible by $M$. Motivated by the parity problem for the partition function, $p(n)$, Gordon and Ono studied the generating functions for $t$-regular partitions, and determined conditions for when these functions are lacunary modulo powers of primes. We generalize their results in a number of ways by studying infinite products called Dedekind eta-quotients and generalized Dedekind eta-quotients. We then apply our results to the generating functions for the partition functions considered by Nekrasov, Ok…
Let $\left(a_{n}\right)_{n=1}^{\infty}$ be a lacunary sequence of positive real numbers. Rudnick and Technau showed that for almost all $α\in\mathbb{R}$, the pair correlation of $\left(αa_{n}\right)_{n=1}^{\infty}$ mod 1 is Poissonian. We show that all higher correlations and hence the nearest-neighbour spacing distribution are Poissonian as well, thereby extending a result of Rudnick and Zaharescu to real-valued sequences.
We estimate the number of possible types degree patterns of $k$-lacunary polynomials of degree $t < p$ which split completely modulo $p$. The result is based on a combination of a bound on the number of zeros of lacunary polynomials with some graph theory arguments.
Ramanujan (and others) proved that the partition function satisfies a number of striking congruences modulo powers of 5, 7 and 11. A number of further congruences were shown by the works of Atkin, O'Brien, and Newman. In this paper we prove that there are infinitely many such congruences for every prime modulus exceeding 3. In addition, we provide a simple criterion guaranteeing the truth of Newman's conjecture for any prime modulus exceeding 3 (recall that Newman's conjecture asserts that the partition function hits every residue class modulo a given integer M infinitely often).
For a positive integer $\ell$, let $b_{\ell}(n)$ denote the number of $\ell$-regular partitions of a nonnegative integer $n$. Motivated by some recent conjectures of Keith and Zanello, we establish infinite families of congruences modulo $2$ for $b_3(n)$ and $b_{21}(n)$. We prove a specific case of a conjecture of Keith and Zanello on self-similarities of $b_3(n)$ modulo $2$. We next prove that the series $\sum_{n=0}^{\infty}b_9(2n+1)q^n$ is lacunary modulo arbitrary powers of $2$. We also prove that the series $\sum_{n=0}^{\infty}b_9(4n)q^n$ is lacunary modulo $2$.
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 some new relations for Newman digit sums respectively different modulos and put some problems. In particular, for the odd prime modulos we put an important conjecture.
Maximal volume of convex bodies avoiding random points and lacunary dilates
Beijing Institute of Mathematical Sciences and Applications (BIMSA)（北京数学科学与应用研究院）；Shanghai Institute for Mathematics and Interdisciplinary Sciences (SIMIS)（上海数学与交叉学科研究院）；Fudan University（复旦大学）
The minimal excludant, or "mex" function, on a set $S$ of positive integers is the least positive integer not in $S$. In a recent paper, Andrews and Newman extended the mex-function to integer partitions and found numerous surprising partition identities connected with these functions. Very recently, da Silva and Sellers present parity considerations of one of the families of functions Andrews and Newman studied, namely $p_{t,t}(n)$, and provide complete parity characterizations of $p_{1,1}(n)$ and $p_{3,3}(n)$. In this article, we study the parity of $p_{t,t}(n)$ when $t=2^α, 3\cdot 2^α$ for …
One of the fundamental theorems of uniform distribution theory states that the fractional parts of the sequence $(n α)_{n \geq 1}$ are uniformly distributed modulo one (u.d. mod 1) for every irrational number $α$. Another important result of Weyl states that for every sequence $(n_k)_{k \geq 1}$ of distinct positive integers the sequence of fractional parts of $(n_k α)_{k \geq 1}$ is u.d. mod 1 for almost all $α$. However, in this general case it is usually extremely difficult to classify those $α$ for which uniform distribution occurs, and to measure the speed of convergence of the empirical …
We prove two generalizations of Furstenberg's Diophantine result regarding density of an orbit of an irrational point in the one-torus under the action of multiplication by a non-lacunary multiplicative semi-group of $\mathbb{N}$. We show that for any sequences $\{a_{n} \},\{b_{n} \}\subset\mathbb{Z}$ for which the quotients of successive elements tend to $1$ as $n$ goes to infinity, and any infinite sequence $\{c_{n} \}$, the set $\{a_{n}b_{m}c_{k}x : n,m,k\in\mathbb{N} \}$ is dense modulo $1$ for every irrational $x$. Moreover, by ergodic-theoretical methods, we prove that if $\{a_{n} \},\{b…
For a positive integer $t\geq 2$, let $b_{t}(n)$ denote the number of $t$-regular partitions of a nonnegative integer $n$. In a recent paper, Keith and Zanello investigated the parity of $b_{t}(n)$ when $t\leq 28$. They discovered new infinite families of Ramanujan type congruences modulo 2 for $b_{21}(n)$ involving every prime $p$ with $p\equiv 13, 17, 19, 23 \pmod{24}$. In this paper, we investigate the parity of $b_{21}(n)$ involving the primes $p$ with $p\equiv 1, 5, 7, 11 \pmod{24}$. We prove new infinite families of Ramanujan type congruences modulo 2 for $b_{21}(n)$ involving the odd pr…
By a classical result of Weyl, for any increasing sequence $(n_k)_{k \geq 1}$ of integers the sequence of fractional parts $(\{n_k x\})_{k \geq 1}$ is uniformly distributed modulo 1 for almost all $x \in [0,1]$. Except for a few special cases, e.g. when $n_k=k, k \geq 1$, the exceptional set cannot be described explicitly. The exact asymptotic order of the discrepancy of $(\{n_k x\})_{k \geq 1}$ is only known in a few special cases, for example when $(n_k)_{k \geq 1}$ is a (Hadamard) lacunary sequence, that is when $n_{k+1}/n_k \geq q > 1, k \geq 1$. In this case of quickly increasing $(n_k)_{…
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…
Andrews and Newman introduced the mex-function $\text{mex}_{A,a}(λ)$ for an integer partition $λ$ of a positive integer $n$ as the smallest positive integer congruent to $a$ modulo $A$ that is not a part of $λ$. They then defined $p_{A,a}(n)$ to be the number of partitions $λ$ of $n$ satisfying $\text{mex}_{A,a}(λ)\equiv a\pmod{2A}$. They found the generating function for $p_{t,t}(n)$ and $p_{2t,t}(n)$ for any positive integer $t$, and studied their arithmetic properties for some small values of $t$. In this article, we study the partition function $p_{mt,t}(n)$ for all positive integers $m$ a…
