arXiv subjects

Verónica Becher

Publications and source records attributed to Verónica Becher.

At least 19 recordsLinked to original sources

arXiv2026

Normal numbers in sparse Cantor sets

We consider Cantor-type sets of Hausdorff dimension zero, consisting of all numbers whose base-2 expansion can have a 1 only at positions belonging to a given sparse set (local count at least log k in every interval of length k). We prove that the measure induced by independent, non-identically distributed Bernoulli digits assigns full mass to numbers that are normal in every odd base. The proof extends Schmidt's 1960 method to this Hausdorff zero-dimensional setting, and we provide an explicit algorithmic construction of such numbers -- yielding the first known examples of numbers deterministic in base~2 yet normal in all odd bases. This work supports our broader conjecture that given determinism in one base, normality in all multiplicatively independent bases is prevalent.

math.NT↗

arXiv2025

Poisson genericity in numeration systems with exponentially mixing probabilities

We define Poisson genericity for infinite sequences in any finite or countable alphabet with an invariant exponentially-mixing probability measure. A sequence is Poisson generic if the number of occurrences of blocks of symbols asymptotically follows a Poisson law as the block length increases. We prove that almost all sequences are Poisson generic. Our result generalizes Peres and Weiss' theorem about Poisson genericity of integral bases numeration systems. In particular, we obtain that their continued fraction expansions for almost all real numbers are Poisson generic.

math.PR↗

arXiv2025

Rauzy dimension and finite-state dimension

In 1976, Rauzy studied two complexity functions, $\underlineβ$ and $\overlineβ$, for infinite sequences over a finite alphabet. The function $\underlineβ$ achieves its maximum precisely for Borel normal sequences, while $\overlineβ$ reaches its minimum for sequences that, when added to any Borel normal sequence, result in another Borel normal sequence. We establish a connection between Rauzy's complexity functions, $\underlineβ$ and $\overlineβ$, and the notions of non-aligned block entropy, $\underline{h}$ and $\overline{h}$, by providing sharp upper and lower bounds for $\underline{h}$ in terms of $\underlineβ$, and sharp upper and lower bounds for $\overline{h}$ in terms of $\overlineβ$. We adopt a probabilistic approach by considering an infinite sequence of random variables over a finite alphabet. The proof relies on a new characterization of non-aligned block entropies, $\overline{h}$ and $\underline{h}$, in terms of Shannon's conditional entropy. The bounds imply that sequences with $\overline{h} = 0$ coincide with those for which $\overlineβ = 0$. We also show that the non-aligned block entropies, $\underline{h}$ and $\overline{h}$, are essentially subadditive.

cs.IT↗

arXiv2025

Automata for the commutative closure of regular sets

Consider $ A^* $, the free monoid generated by the finite alphabet $A$ with the concatenation operation. Two words have the same commutative image when one is a permutation of the symbols of the other. The commutative closure of a set $ L \subseteq A^* $ is the set $ {C}(L) \subseteq A^* $ of words whose commutative image coincides with that of some word in $ L $. We provide an algorithm that, given a regular set $ L $, produces a finite state automaton that accepts the commutative closure $ {C}(L) $, provided that this closure set is regular. The problem of deciding whether $ {C}(L) $ is regular was solved by Ginsburg and Spanier in 1966 using the decidability of Presburger sentences, and by Gohon in 1985 via formal power series. The problem of constructing an automaton that accepts $ {C}(L) $ has already been studied in the literature. We give a simpler algorithm using an algebraic approach.

cs.FL↗

arXiv2024

Lyndon pairs and the lexicographically greatest perfect necklace

Fix a finite alphabet. A necklace is a circular word. For positive integers $n$ and~$k$, a necklace is $(n,k)$-perfect if all words of length $n$ occur $k$ times but at positions with different congruence modulo $k$, for any convention of the starting position. We define the notion of a Lyndon pair and we use it to construct the lexicographically greatest $(n,k)$-perfect necklace, for any $n$ and $k$ such that $n$ divides~$k$ or $k$ divides~$n$. Our construction generalizes Fredricksen and Maiorana's construction of the lexicographically greatest de Bruijn sequence of order $n$, based on the concatenation of the Lyndon words whose length divide $n$.

math.CO↗

arXiv2024

De Bruijn Sequences with Minimum Discrepancy

The discrepancy of a binary string is the maximum (absolute) difference between the number of ones and the number of zeroes over all possible substrings of the given binary string. In this note we determine the minimal discrepancy that a binary de Bruijn sequence of order $n$ can achieve, which is $n$. This was an open problem until now. We give an algorithm that constructs a binary de Bruijn sequence with minimal discrepancy. A slight modification of this algorithm deals with arbitrary alphabets and yields de Bruijn sequences of order $n$ with discrepancy at most $1$ above the trivial lower bound $n$.

cs.DM↗

arXiv2024

The discrepancy of the Champernowne constant

A number is normal in base $b$ if, in its base $b$ expansion, all blocks of digits of equal length have the same asymptotic frequency. The rate at which a number approaches normality is quantified by the classical notion of discrepancy, which indicates how far the scaling of the number by powers of $b$ is from being equidistributed modulo 1. This rate is known as the discrepancy of a normal number. The Champernowne constant $c_{10} = 0.12345678910111213141516\ldots$ is the most well-known example of a normal number. In 1986, Schiffer provided the discrepancy of numbers in a family that includes the Champernowne constant. His proof relies on exponential sums. Here, we present a discrete and elementary proof specifically for the discrepancy of the Champernowne constant.

math.NT↗

arXiv2024

A construction of a $λ$- Poisson generic sequence

Years ago Zeev Rudnick defined the $λ$-Poisson generic sequences as the infinite sequences of symbols in a finite alphabet where the number of occurrences of long words in the initial segments follow the Poisson distribution with parameter $λ$. Although almost all sequences, with respect to the uniform measure, are Poisson generic, no explicit instance has yet been given. In this note we give a construction of an explicit $λ$-Poisson generic sequence over any alphabet and any positive $λ$, except for the case of the two-symbol alphabet, in which it is required that $λ$ be less than or equal to the natural logarithm of $2$. Since $λ$-Poisson genericity implies Borel normality, the constructed sequences are Borel normal. The same construction provides explicit instances of Borel normal sequences that are not $λ$-Poisson generic.

math.NT↗

arXiv2023

On extremal factors of de Bruijn-like graphs

In 1972 Mykkeltveit proved that the maximum number of vertex-disjoint cycles in the de Bruijn graphs of order $n$ is attained by the pure cycling register rule, as conjectured by Golomb. We generalize this result to the tensor product of the de Bruijn graph of order $n$ and a simple cycle of size $k$, when $n$ divides $k$ or vice versa. We also develop counting formulae for a large family of cycling register rules, including the linear register rules proposed by Golomb.

math.CO↗

arXiv2023

On simply normal numbers with digit dependencies

Given an integer $b\geqslant 2$ and a set $P$ of prime numbers, the set $T_P $ of Toeplitz numbers comprises all elements of $[0,b[$ whose digits $(a_n)_{n\geqslant 1}$ in the base-$b$ expansion satisfy $a_n=a_{pn}$ for all $p\in P$ and $n\geqslant 1$. Using a completely additive arithmetical function, we construct a number in~$T_P$ that is simply Borel normal if, and only if, $\sum_{p\not \in P} 1/p=\infty$. We then provide an effective bound for the discrepancy.

math.NT↗

arXiv2023

The descriptive complexity of the set of Poisson generic numbers

Let $b\ge 2$ be an integer. We show that the set of real numbers that are Poisson generic in base $b$ is $\boldsymbolΠ^0_3$-complete in the Borel hierarchy of subsets of the real line. Furthermore, the set of real numbers that are Borel normal in base $b$ and not Poisson generic in base $b$ is complete for the class given by the differences between $\boldsymbolΠ^0_3$ sets. We also show that the effective versions of these results hold in the effective Borel hierarchy.

math.LO↗

arXiv2023

Nested perfect toroidal arrays

We introduce two-dimensional toroidal arrays that are a variant of the de Bruijn tori. We call them nested perfect toroidal arrays. Instead of asking that every array of a given size has exactly one occurrence, we partition the positions in congruence classes and we ask exactly one occurrence in each congruence class. We also ask that this property applies recursively to each of the subarrays. We give a method to construct nested perfect toroidal arrays based on Pascal triangle matrix modulo 2. For the two-symbol alphabet, and for $n$ being a power of $2$, our method yields $2^{n^2+n-1}$ different nested perfect toroidal arrays allocating all the different $n\times n$ arrays in each congruence class that arises from taking the line number modulo $n$ and the column number modulo $n$.

cs.IT↗

arXiv2022

Poisson generic sequences

Years ago, Zeev Rudnick defined the Poisson generic real numbers by counting the number of occurrences of long blocks of digits in the initial segments of the expansions of the real numbers in a fixed integer base. Peres and Weiss proved that almost all real numbers, with respect to Lebesgue measure, are Poisson generic, but they did not publish their proof. In this note first we transcribe Peres and Weiss' proof and then we show that there are computable Poisson generic instances and that all Martin-Löf random real numbers are Poisson generic.

math.NT↗

arXiv2022

On the number of words with restrictions on the number of symbols

We show that, in an alphabet of $n$ symbols, the number of words of length $n$ whose number of different symbols is away from $(1-1/e)n$, which is the value expected by the Poisson distribution, has exponential decay in $n$. We use Laplace's method for sums and known bounds of Stirling numbers of the second kind. We express our result in terms of inequalities.

math.CO↗

arXiv2021

Randomness and uniform distribution modulo one

We elaborate the notions of Martin-Löf and Schnorr randomness for real numbers in terms of uniform distribution of sequences. We give a necessary condition for a real number to be Schnorr random expressed in terms of classical uniform distribution of sequences. This extends the result proved by Avigad for sequences of linear functions with integer coefficients to the wider classical class of Koksma sequences of functions. And, by requiring equidistribution with respect to every computably enumerable open set (respectively, computably enumerable open set with computable measure) in the unit interval, we give a sufficient condition for Martin-Löf (respectively Schnorr) randomness.

math.LO↗

arXiv2021

Insertion in constructed normal numbers

Defined by Borel, a real number is normal to an integer base $b$, greater than or equal to $2$, if in its base-$b$ expansion every block of digits occurs with the same limiting frequency as every other block of the same length. We consider the problem of insertion in constructed base-$b$ normal expansions to obtain normality to base $(b+1)$.

math.NT↗

arXiv2021

On a question of Mendès France on normal numbers

In 2008 or earlier, Michel Mendès France asked for an instance of a real number $x$ such that both $x$ and $1/x$ are simply normal to a given integer base $b$. We give a positive answer to this question by constructing a number $x$ such that both $x$ and its reciprocal $1/x$ are continued fraction normal as well as normal to all integer bases greater than or equal to $2$. Moreover, $x$ and $1/x$ are both computable.

math.NT↗

arXiv2020

Extending de Bruijn sequences to larger alphabets

A de Bruijn sequence of order n over a k-symbol alphabet is a circular sequence where each length-n sequence occurs exactly once. We present a way of extending de Bruijn sequences by adding a new symbol to the alphabet: the extension is performed by embedding a given de Bruijn sequence into another one of the same order, but over the alphabet with one more symbol, while ensuring that there are no long runs without the new symbol. Our solution is based on auxiliary graphs derived from the de Bruijn graph and solving a problem of maximum flow.

cs.DM↗