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Jakub Byszewski

Publications and source records attributed to Jakub Byszewski.

At least 19 recordsLinked to original sources
arXiv2024

Dynamics of endomorphisms of algebraic groups

Let $σ$ denote an endomorphism of a smooth algebraic group $G$ over the algebraic closure of a finite field, and assume all iterates of $σ$ have finitely many fixed points. Steinberg gave a formula for the number of fixed points of $σ$ (and hence of all of its iterates $σ^n$) in the semisimple case, leading to a representation of its Artin-Mazur zeta function as a rational function. We generalise this to an arbitrary (smooth) algebraic group $G$, where the number of fixed points $σ_n$ of $σ^n$ can depend on $p$-adic properties of $n$. We axiomatise the structure of the sequence $(σ_n)$ via the concept of a `finite-adelically distorted' (FAD-)sequence. Such sequences also occur in topological dynamics, and our subsequent results about zeta functions and asymptotic counting of orbits apply equally well in that situation; for example, to $S$-integer dynamical systems, additive cellular automata and other compact abelian groups. We prove dichotomies for the associated Artin-Mazur zeta function, and study the analogue of the Prime Number Theorem for the function counting periodic orbits of length $\leq N$. For an algebraic group $G$ we express the error term via the $\ell$-adic cohomological zeta function of $G$.

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arXiv2024

Multiband linear cellular automata and endomorphisms of algebraic vector groups

We propose a correspondence between certain multiband linear cellular automata - models of computation widely used in the description of physical phenomena - and endomorphisms of certain algebraic unipotent groups over finite fields. The correspondence is based on the construction of a universal element specialising to a normal generator for any finite field. We use this correspondence to deduce new results concerning the temporal dynamics of such automata, using our prior, purely algebraic, study of the endomorphism ring of vector groups. These produce 'for free' a formula for the number of fixed points of the $n$-iterate in terms of the $p$-adic valuation of $n$, a dichotomy for the Artin-Mazur dynamical zeta function, and an asymptotic formula for the number of periodic orbits. Since multiband linear cellular automata simulate higher order linear automata (in which states depend on finitely many prior temporal states, not just the direct predecessor), the results apply equally well to that class.

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arXiv2023

Gowers norms for automatic sequences

We show that any automatic sequence can be separated into a structured part and a Gowers uniform part in a way that is considerably more efficient than guaranteed by the Arithmetic Regularity Lemma. For sequences produced by strongly connected and prolongable automata, the structured part is rationally almost periodic, while for general sequences the description is marginally more complicated. In particular, we show that all automatic sequences orthogonal to periodic sequences are Gowers uniform. As an application, we obtain for any $l \geq 2$ and any automatic set $A \subset \mathbb{N}_0$ lower bounds on the number of $l$-term arithmetic progressions - contained in $A$ - with a given difference. The analogous result is false for general subsets of $\mathbb{N}_0$ and progressions of length $\geq 5$.

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arXiv2023

Pisot numbers, Salem numbers, and generalised polynomials

We study sets of integers that can be defined by the vanishing of a generalised polynomial expression. We show that this includes sets of values of linear recurrent sequences of Salem type and some linear recurrent sequences of Pisot type. To this end, we introduce the notion of a generalised polynomial on a number field. We establish a connection between the existence of generalised polynomial expressions for sets of values of linear recurrent sequences and for subsemigroups of multiplicative groups of number fields.

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arXiv2021

Rado's theorem for rings and modules

We extend classical results of Rado on partition regularity of systems of linear equations with integer coefficients to the case when the coefficient ring is either an arbitrary integral domain or a noetherian ring. In particular, we show that a system of homogeneous linear equations over an infinite integral domain is partition regular if and only if the corresponding matrix satisfies the columns conditions. The crucial idea is to study partition regularity for general modules rather than only for rings. Contrary to previous techniques, our approach is independent of the characteristic of the coefficient ring.

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arXiv2020

Automata and finite order elements in the Nottingham group

The Nottingham group at 2 is the group of (formal) power series $t+a_2 t^2+ a_3 t^3+ \cdots$ in the variable $t$ with coefficients $a_i$ from the field with two elements, where the group operation is given by composition of power series. The depth of such a series is the largest $d\geq 1$ for which $a_2=\dots=a_d=0$. Only a handful of power series of finite order are explicitly known through a formula for their coefficients. We argue in this paper that it is advantageous to describe such series in closed computational form through automata, based on effective versions of proofs of Christol's theorem identifying algebraic and automatic series. Up to conjugation, there are only finitely many series $σ$ of order $2^n$ with fixed break sequence (i.e. the sequence of depths of $σ^{\circ 2^i}$). Starting from Witt vector or Carlitz module constructions, we give an explicit automaton-theoretic description of: (a) representatives up to conjugation for all series of order 4 with break sequence (1,m) for m<10; (b) representatives up to conjugation for all series of order 8 with minimal break sequence (1,3,11); and (c) an embedding of the Klein four-group into the Nottingham group at 2. We study the complexity of the new examples from the algebro-geometric properties of the equations they satisfy. For this, we generalise the theory of sparseness of power series to a four-step hierarchy of complexity, for which we give both Galois-theoretic and combinatorial descriptions. We identify where our different series fit into this hierarchy. We construct sparse representatives for the conjugacy class of elements of order two and depth $2^μ\pm 1$ $(μ\geq 1)$. Series with small state complexity can end up high in the hierarchy. This is true, for example, for a new automaton we found, representing a series of order 4 with 5 states, the minimal possible number for such a series.

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arXiv2020

Substitutive systems and a finitary version of Cobham's theorem

We study substitutive systems generated by nonprimitive substitutions and show that transitive subsystems of substitutive systems are substitutive. As an application we obtain a complete characterisation of the sets of words that can appear as common factors of two automatic sequences defined over multiplicatively independent bases. This generalises the famous theorem of Cobham.

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arXiv2019

Dynamically affine maps in positive characteristic

We study fixed points of iterates of dynamically affine maps (a generalisation of Lattès maps) over algebraically closed fields of positive characteristic $p$. We present and study certain hypotheses that imply a dichotomy for the Artin-Mazur zeta function of the dynamical system: it is either rational or non-holonomic, depending on specific characteristics of the map. We also study the algebraicity of the so-called tame zeta function, the generating function for periodic points of order coprime to $p$. We then verify these hypotheses for dynamically affine maps on the projective line, generalising previous work of Bridy, and, in arbitrary dimension, for maps on Kummer varieties arising from multiplication by integers on abelian varieties.

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arXiv2018

Dynamics on abelian varieties in positive characteristic

We study periodic points for endomorphisms $σ$ of abelian varieties $A$ over algebraically closed fields of positive characteristic $p$. We show that the dynamical zeta function $ζ_σ$ of $σ$ is either rational or transcendental, the first case happening precisely when $σ^n-1$ is a separable isogeny for all $n$. We call this condition very inseparability and show it is equivalent to the action of $σ$ on the local $p$-torsion group scheme being nilpotent. The "false" zeta function $D_σ$, in which the number of fixed points of $σ^n$ is replaced by the degree of $σ^n-1$, is always a rational function. Let $1/Λ$ denote its largest real pole and assume no other pole or zero has the same absolute value. Then, using a general dichotomy result for power series proven by Royals and Ward in the appendix, we find that $ζ_σ(z)$ has a natural boundary at $|z|=1/Λ$ when $σ$ is not very inseparable. We introduce and study tame dynamics, ignoring orbits whose order is divisible by $p$. We construct a tame zeta function $ζ^*_σ$ that is always algebraic, and such that $ζ_σ$ factors into an infinite product of tame zeta functions. We briefly discuss functional equations. Finally, we study the length distribution of orbits and tame orbits. Orbits of very inseparable endomorphisms distribute like those of Axiom A systems with entropy $\log Λ$, but the orbit length distribution of not very inseparable endomorphisms is more erratic and similar to $S$-integer dynamical systems. We provide an expression for the prime orbit counting function in which the error term displays a power saving depending on the largest real part of a zero of $D_σ(Λ^{-s})$.

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arXiv2018

Factors of generalised polynomials and automatic sequences

The aim of this short note is to generalise the result of Rampersad--Shallit saying that an automatic sequence and a Sturmian sequence cannot have arbitrarily long common factors. We show that the same result holds if a Sturmian sequence is replaced by an arbitrary sequence whose terms are given by a generalised polynomial (i.e., an expression involving algebraic operations and the floor function) that is not periodic except for a set of density zero.

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arXiv2017

Automatic sequences and generalised polynomials

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 conjecture, proving that any hypothetical counterexample is periodic away from a very sparse and structured set. In particular, we show that for a polynomial $p(n)$ with at least one irrational coefficient (except for the constant one) and integer $m\geq 2$, the sequence $\lfloor p(n) \rfloor \bmod{m}$ is never automatic. We also prove that the conjecture is equivalent to the claim that the set of powers of an integer $k\geq 2$ is not given by a generalised polynomial.

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arXiv2017

A density version of Cobham's theorem

Cobham's theorem asserts that if a sequence is automatic with respect to two multiplicatively independent bases, then it is ultimately periodic. We prove a stronger density version of the result: if two sequences which are automatic with respect to two multiplicatively independent bases coincide on a set of density one, then they also coincide on a set of density one with a periodic sequence. We apply the result to a problem of Deshouillers and Ruzsa concerning the least nonzero digit of $n!$ in base $12$.

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arXiv2016

Sparse generalised polynomials

We investigate generalised polynomials (i.e. polynomial-like expressions involving the use of the floor function) which take the value $0$ on all integers except for a set of density $0$. Our main result is that the set of integers where a sparse generalised polynomial takes non-zero value cannot contain a translate of an IP set. We also study some explicit constructions, and show that the characteristic functions of the Fibonacci and Tribonacci numbers are given by generalised polynomails. Finally, we show that any sufficiently sparse $\{0,1\}$-valued sequence is given by a generalised polynomial. (This paper is essentially the first half of our earlier submission arXiv:1610.03900 [math.NT]. Because the material in arXiv:1610.03900 [math.NT] touches upon many different subjects, we believe it is preferable to split it into two independent papers.)

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arXiv2016

Simultaneous $p$-orderings and minimising volumes in number fields

In the paper "On the interpolation of integer-valued polynomials" (Journal of Number Theory 133 (2013), pp. 4224--4232.) V. Volkov and F. Petrov consider the problem of existence of the so-called $n$-universal sets (related to simultaneous $p$-orderings of Bhargava) in the ring of Gaussian integers. We extend their results to arbitrary imaginary quadratic number fields and prove an existence theorem that provides a strong counterexample to a conjecture of Volkov-Petrov on minimal cardinality of $n$-universal sets. Along the way, we discover a link with Euler-Kronecker constants and prove a lower bound on Euler-Kronecker constants which is of the same order of magnitude as the one obtained by Ihara.

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arXiv2016

Automatic sequences, generalised polynomials, and nilmanifolds

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 resolve this conjecture, proving that any hypothetical counterexample is periodic away from a very sparse and structured set. In particular, we show that for a polynomial $p(n)$ with at least one irrational coefficient (except for the constant one) and integer $m$, the sequence $\lfloor p(n) \rfloor \bmod{m}$ is never automatic. We also obtain a conditional result, where we prove the conjecture under the assumption that the characteristic sequence of the set of powers of an integer $k\geq 2$ is not given by a generalised polynomial.

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arXiv2015

Transitive dendrite map with zero entropy

Hoehn and Mouron [Ergod. Th. \& Dynam. Sys. (2014) \textbf{34}, 1897--1913] constructed a map on the universal dendrite that is topologically weakly mixing but not mixing. We modify the Hoehn-Mouron example to show that there exists a transitive (even weakly mixing) dendrite map with zero topological entropy. This answers the question of Baldwin [Topology (2001) \textbf{40}, 551--569].

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