Multiplicative automatic sequences
We obtain a complete classification of complex-valued sequences which are both multiplicative and automatic.
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arXiv subjects
Jakub Konieczny
Publications and source records attributed to Jakub Konieczny.
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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On uniformity of $q$-multiplicative sequences
We show that any $q$-multiplicative sequence which is \emph{oscillating} of order $1$, i.e.\ does not correlate with linear phase functions $e^{2πi nα}$ ($α\in \mathbb{R})$, is Gowers uniform of all orders, and hence in particular does not correlate with polynomial phase functions $e^{2πi p(n)}$ ($p \in \mathbb{R}[x]$). Quantitatively, we show that any $q$-multiplicative sequence which is of Gelfond type of order 1 is automatically of Gelfond type of all orders. Consequently, any such $q$-multiplicative sequence is a good weight for ergodic theorems. We also obtain combinatorial corollaries concerning linear patterns in sets which are described in terms of sums of digits.
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Möbius orthogonality for $q$-semimultiplicative sequences
We show that all $q$-semimultiplicative sequences are asymptotically orthogonal to the Möbius function, thus proving the Sarnak conjecture for this class of sequences. This generalises analogous results for the sum-of-digits function and other digital sequences which follow from previous work of Mauduit and Rivat.
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Sets of recurrence as bases for the positive integers
We study sets of the form $A = \big\{ n \in \mathbb N \big| \lVert p(n) \rVert_{\mathbb R / \mathbb Z} \leq \varepsilon(n) \big\}$ for various real valued polynomials $p$ and decay rates $\varepsilon$. In particular, we ask when such sets are bases of finite order for the positive integers. We show that generically, $A$ is a basis of order $2$ when $\operatorname{deg} p \geq 3$, but not when $\operatorname{deg} p = 2$, although then $A + A$ still has asymptotic density $1$.
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Combinatorial properties of Nil-Bohr sets
In this paper we study the relation between two notions of largeness that apply to a set of positive integers, namely $\mathrm{Nil}_d{-}\mathrm{Bohr}$ and $\mathrm{SG}_k$, as introduced by Host and Kra. We prove that any $\mathrm{Nil}_d{-}\mathrm{Bohr}_0$ set is necessarily $\mathrm{SG}_k$ where ${k}$ is effectively bounded in terms of $d$. This partially resolves a conjecture of Host and Kra.
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Weakly mixing sets and polynomial equations
We investigate polynomial patterns which can be guaranteed to appear in \emph{weakly mixing} sets introduced by introduced by Furstenberg and studied by Fish. In particular, we prove that if $A \subset \mathbb N$ is a weakly mixing set and $p(x) \in \mathbb Z[x]$ a polynomial of odd degree with positive leading coefficient, then all sufficiently large integers $N$ can be represented as $N = n_1 + n_2$, where $p(n_1) + m,\ p(n_2) + m \in A$ for some $m \in A$.
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Automatic sequences as good weights for ergodic theorems
We study correlation estimates of automatic sequences (that is, sequences computable by finite automata) with polynomial phases. As a consequence, we provide a new class of good weights for classical and polynomial ergodic theorems, not coming themselves from dynamical systems. We show that automatic sequences are good weights in $L^2$ for polynomial averages and totally ergodic systems. For totally balanced automatic sequences (i.e., sequences converging to zero in mean along arithmetic progressions) the pointwise weighted ergodic theorem in $L^1$ holds. Moreover, invertible automatic sequences are good weights for the pointwise polynomial ergodic theorem in $L^r$, $r>1$.
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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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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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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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Gowers norms for the Thue-Morse and Rudin-Shapiro sequences
We estimate Gowers uniformity norms for some classical automatic sequences, such as the Thue-Morse and Rudin-Shapiro sequences. The methods can also be extended to other automatic sequences. As an application, we asymptotically count arithmetic progressions in the set of integers $\leq N$ where the Thue-Morse (resp. Rudin-Shapiro) sequence takes the value $+1$.
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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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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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Arcwise connectedness of the set of ergodic measures of hereditary shifts
We show that the set of ergodic invariant measures of a shift space with a safe symbol (this includes all hereditary shifts) is arcwise connected when endowed with the $d$-bar metric. As a consequence the set of ergodic measures of such a shift is also arcwise connected in the weak-star topology and the entropy function over this set attains all values in the interval between zero and the topological entropy of the shift (inclusive). The latter result is motivated by a conjecture of A.~Katok.
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Applications of Ultrafilters in Ergodic Theory and Combinatorial Number Theory
Ultrafilters are very useful and versatile objects with applications throughout mathematics: in topology, analysis, combinarotics, model theory, and even theory of social choice. Proofs based on ultrafilters tend to be shorter and more elegant than their classical counterparts. In this thesis, we survey some of the most striking ways in which ultrafilters can be exploited in combinatorics and ergodic theory, with a brief mention of model theory. In the initial sections, we establish the basics of the theory of ultrafilters in the hope of keeping our exposition possibly self-contained, and then proceed to specific applications. Important combinatorial results we discuss are the theorems of Hindman, van der Waerden and Hales-Jewett. Each of them asserts essentially that in a finite partition of, respectively, the natural numbers or words over a finite alphabet, one cell much of the combinatorial structure. We next turn to results in ergodic theory, which rely strongly on combinatorial preliminaries. They assert essentially that certain sets of return times are combinatorially rich. We finish by presenting the ultrafilter proof of the famous Arrow's Impossibility Theorem and the construction of the ultraproduct in model theory.
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