arXiv subjects

Daniel Dombek

Publications and source records attributed to Daniel Dombek.

5 recordsLinked to original sources

arXiv2025

On biquadratic fields: when 5 squares are not enough

In this paper we study the Pythagoras number $\mathcal{P}(\mathcal{O}_K)$ for the rings of integers in totally real biquadratic fields $K$. We continue the work of Tinkov\'a towards proving the conjecture by Kr\'asensk\'y, Ra\v{s}ka and Sgallov\'a that a biquadratic $K$ satisfies $\mathcal{P}(\mathcal{O}_K)\geq 6$ if and only if it contains neither $\sqrt{2}$ nor $\sqrt{5}$, with only finitely many exceptions. We fully solve two out of three remaining classes of fields by proving that all but finitely many $K$ containing $\sqrt{6}$ or $\sqrt{7}$ satisfy $\mathcal{P}(\mathcal{O}_K)\geq 6$. Furthermore, we present ideas and computations which further support the conjecture also for $K$ containing $\sqrt{3}$. This enables us to refine the conjecture by explicitly listing the exceptional fields.

math.NT↗

arXiv2014

On distinct unit generated fields that are totally complex

We consider the problem of characterizing all number fields $K$ such that all algebraic integers $α\in K$ can be written as the sum of distinct units of $K$. We extend a method due to Thuswaldner and Ziegler that previously did not work for totally complex fields and apply our results to the case of totally complex quartic number fields.

math.NT↗

arXiv2014

Confluent Parry numbers, their spectra, and integers in positive- and negative-base number systems

In this paper we study the expansions of real numbers in positive and negative real base as introduced by Rényi, and Ito & Sadahiro, respectively. In particular, we compare the sets $\mathbb{Z}_β^+$ and $\mathbb{Z}_{-β}$ of nonnegative $β$-integers and $(-β)$-integers. We describe all bases $(\pmβ)$ for which $\mathbb{Z}_β^+$ and $\mathbb{Z}_{-β}$ can be coded by infinite words which are fixed points of conjugated morphisms, and consequently have the same language. Moreover, we prove that this happens precisely for $β$ with another interesting property, namely that any integer linear combination of non-negative powers of the base $-β$ with coefficients in $\{0,1,\dots,\lfloorβ\rfloor\}$ is a $(-β)$-integer, although the corresponding sequence of digits is forbidden as a $(-β)$-integer.

math.CO↗

arXiv2011

Substitutions over infinite alphabet generating (-β)-integers

This contribution is devoted to the study of positional numeration systems with negative base introduced by Ito and Sadahiro in 2009, called (-β)-expansions. We give an admissibility criterion for more general case of (-β)-expansions and discuss the properties of the set of (-β)-integers. We give a description of distances within this set and show that this set can be coded by an infinite word over an infinite alphabet, which is a fixed point of a non-erasing non-trivial morphism.

cs.DM↗

arXiv2011

Number representation using generalized $(-β)$-transformation

We study non-standard number systems with negative base $-β$. Instead of the Ito-Sadahiro definition, based on the transformation $T_{-β}$ of the interval $\big[-\fracβ{β+1},\frac{1}{β+1}\big)$ into itself, we suggest a generalization using an interval $[l,l+1)$ with $l\in(-1,0]$. Such generalization may eliminate certain disadvantages of the Ito-Sadahiro system. We focus on the description of admissible digit strings and their periodicity.

cs.DM↗