arXiv subjects

Tomáš Hejda

Publications and source records attributed to Tomáš Hejda.

8 recordsLinked to original sources

arXiv2021

Ternary quadratic forms representing a given arithmetic progression

A positive quadratic form is $(k,\ell)$-universal if it represents all the numbers $kx+\ell$ where $x$ is a non-negative integer, and almost $(k,\ell)$-universal if it represents all but finitely many of them. We prove that for any $k,\ell$ such that $k\nmid\ell$ there exists an almost $(k,\ell)$-universal diagonal ternary form. We also conjecture that there are only finitely many primes $p$ for which a $(p,\ell)$-universal diagonal ternary form exists (for any $\ell<p$) and we show the results of computer experiments that speak in favor of the conjecture.

math.NT↗

arXiv2019

Additive structure of totally positive quadratic integers

Let $K=\mathbb Q(\sqrt D)$ be a real quadratic field. We consider the additive semigroup $\mathcal O_K^+(+)$ of totally positive integers in $K$ and determine its generators (indecomposable integers) and relations; they can be nicely described in terms of the periodic continued fraction for $\sqrt D$. We also characterize all uniquely decomposable integers in $K$ and estimate their norms. Using these results, we prove that the semigroup $\mathcal O_K^+(+)$ completely determines the real quadratic field $K$.

math.NT↗

arXiv2018

Multiple tilings associated to d-Bonacci beta-expansions

Let $β\in(1,2)$ be a Pisot unit and consider the symmetric $β$-expansions. We give a necessary and sufficient condition for the associated Rauzy fractals to form a tiling of the contractive hyperplane. For $β$ a $d$-Bonacci number, i.e., Pisot root of $x^d-x^{d-1}-\dots-x-1$ we show that the Rauzy fractals form a multiple tiling with covering degree $d-1$.

math.NT↗

arXiv2014

Beta-expansions of rational numbers in quadratic Pisot bases

We study rational numbers with purely periodic Rényi $β$-expansions. For bases $β$ satisfying $β^2=aβ+b$ with $b$ dividing $a$, we give a necessary and sufficient condition for $γ(β)=1$, i.e., that all rational numbers $p/q\in[0,1)$ with $\gcd(q,b)=1$ have a purely periodic $β$-expansion. A simple algorithm for determining the value of $γ(β)$ for all quadratic Pisot numbers $β$ is described.

math.DS↗

arXiv2014

Spectral properties of cubic complex Pisot units

For a real number $β>1$, Erdős, Joó and Komornik study distances between consecutive points in the set $X^m(β)=\bigl\{\sum_{j=0}^n a_j β^j : n\in\mathbb N,\,a_j\in\{0,1,\dots,m\}\bigr\}$. Pisot numbers play a crucial role for the properties of $X^m(β)$. Following the work of Zaïmi, who considered $X^m(γ)$ with $γ\in\mathbb{C}\setminus\mathbb{R}$ and $|γ|>1$, we show that for any non-real $γ$ and $m < |γ|^2-1$, the set $X^m(γ)$ is not relatively dense in the complex plane. Then we focus on complex Pisot units with a positive real conjugate $γ'$ and $m > |γ|^2-1$. If the number $1/γ'$ satisfies Property (F), we deduce that $X^m(γ)$ is uniformly discrete and relatively dense, i.e., $X^m(γ)$ is a Delone set. Moreover, we present an algorithm for determining two parameters of the Delone set $X^m(γ)$ which are analogous to minimal and maximal distances in the real case $X^m(β)$. For $γ$ satisfying $γ^3 + γ^2 + γ- 1 = 0$, explicit formulas for the two parameters are given.

math.MG↗

arXiv2013

Balancedness of Arnoux-Rauzy and Brun words

We study balancedness properties of words given by the Arnoux-Rauzy and Brun multi-dimensional continued fraction algorithms. We show that almost all Brun words on 3 letters and Arnoux-Rauzy words over arbitrary alphabets are finitely balanced; in particular, boundedness of the strong partial quotients implies balancedness. On the other hand, we provide examples of unbalanced Brun words on 3 letters.

cs.FL↗

arXiv2013

Greedy and lazy representations in negative base systems

We consider positional numeration systems with negative real base $-β$, where $β>1$, and study the extremal representations in these systems, called here the greedy and lazy representations. We give algorithms for determination of minimal and maximal $(-β)$-representation with respect to the alternate order. We also show that both extremal representations can be obtained as representations in the positive base $β^2$ and a non-integer alphabet. This enables us to characterize digit sequences admissible as greedy and lazy $(-β)$-representation. Such a characterization allows us to study the set of uniquely representable numbers. In case that $β$ is the golden ratio and the Tribonacci constant, we give the characterization of digit sequences admissible as greedy and lazy $(-β)$-representation using a set of forbidden strings.

math.DS↗

arXiv2012

Morphisms preserving the set of words coding three interval exchange

Any amicable pair ϕ, ψ of Sturmian morphisms enables a construction of a ternary morphism η which preserves the set of infinite words coding 3-interval exchange. We determine the number of amicable pairs with the same incidence matrix in $SL^\pm(2,N)$ and we study incidence matrices associated with the corresponding ternary morphisms η.

math.CO↗