Related papers: Automatic congruences for diagonals of rational fu…
Abstract numeration systems encode natural numbers using radix ordered words of an infinite regular language and linear recurrence sequences play a key role in their valuation. Sequence automata, which are deterministic finite automata with…
We conjecture that bounded generalised polynomial functions cannot be generated by finite automata, except for the trivial case when they are periodic away from a finite set. Using methods from ergodic theory, we are able to partially…
We prove decidability results on the existence of constant subsequences of uniformly recurrent morphic sequences along arithmetic progressions. We use spectral properties of the subshifts they generate to give a first algorithm deciding…
Let $D$ be an open disk of radius $\le 1$ in $\mathbb C$, and let $(\epsilon_n)$ be a sequence of $\pm 1$. We prove that for every analytic function $f: D \to \mathbb C$ without zeros in $D$, there exists a unique sequence $(\alpha_n)$ of…
In the paper we develop the $p$-adic theory of discrete automata. Every automaton $\mathfrak A$ (transducer) whose input/output alphabets consist of $p$ symbols can be associated to a continuous (in fact, 1-Lipschitz) map from $p$-adic…
We obtain a complete classification of complex-valued sequences which are both multiplicative and automatic.
We conjecture that bounded generalised polynomial functions cannot be generated by finite automata, except for the trivial case when they are ultimately periodic. Using methods from ergodic theory, we are able to partially resolve this…
We prove a quantitative version of a result of Furstenberg and Deligne stating that the the diagonal of a multivariate algebraic power series with coefficients in a field of positive characteristic is algebraic. As a consequence, we obtain…
We study the $K$-Fibonacci sequence $\mathcal{F}_p$ modulo prime $p$. Cardinalities of sets $|\mathcal{F}_p+\mathcal{F}_p|$ and $|\mathcal{F}_p\cdot\mathcal{F}_p|$ are estimated. We present the method of estimating doubling constant of some…
We present a method for obtaining congruences modulo powers of 3 for sequences given by recurrences of finite depth with polynomial coefficients. We apply this method to Catalan numbers, Motzkin numbers, Riordan numbers, Schr\"oder numbers,…
The paper constructs an infinite sequence of strongly regular directed graphs. The construction is based on representing adjacency matrices as block matrices composed of circulant blocks, together with the use of a compactification…
For each of the $15$ known sporadic Ap\'ery-like sequences, we prove congruences modulo $p^2$ that are natural extensions of the Lucas congruences modulo $p$. This extends a result of Gessel for the numbers used by Ap\'ery in his proof of…
We study those automatic sequences which are produced by an automaton whose underlying graph is the Cayley graph of a finite group. For $2$-automatic sequences, we find a characterization in terms of what we call homogeneity, and among…
In this work, we prove that many Ap\'ery-like sequences arising from modular forms satisfy the Lucas congruences modulo any prime. As an implication, we completely affirm four conjectural Lucas congruences that were recently posed by S.…
Let $p_k(n)$ be given by the $k$-th power of the Euler Product $\prod _{n=1}^{\infty}(1-q^n)^k=\sum_{n=0}^{\infty}p_k(n)q^{n}$. By investigating the properties of the modular equations of the second and the third order under the Atkin…
In 1984, Deligne proved that for any prime number $p$, the reduction modulo $p$ of the diagonal of a multivariate algebraic power series with integer coefficients is algebraic over the field of rational functions with coefficients in…
We develop a meta-algorithm that, given a polynomial (in one or more variables), and a prime p, produces a fast (logarithmic time) algorithm that takes a positive integer n and outputs the number of times each residue class modulo p appears…
Many famous integer sequences including the Catalan numbers and the Motzkin numbers can be expressed in the form $ConstantTermOf\left[P(x)^nQ(x)\right]$ for Laurent polynomials $Q$, and symmetric Laurent trinomials $P$. In this paper we…
We give bounds for the number and the size of the primes $p$ such that a reduction modulo $p$ of a system of multivariate polynomials over the integers with a finite number $T$ of complex zeros, does not have exactly $T$ zeros over the…
We propose to associate to a modular form (an infinite number of) complex valued functions on the $p$-adic numbers $\mathbb{Q}_p$ for each prime $p$. We elaborate on the correspondence and study its consequence in terms of the Mellin…