arXiv subjects

Thomas Stoll

Publications and source records attributed to Thomas Stoll.

13 recordsLinked to original sources

arXiv2022

On the binary digits of $n$ and $n^2$

Let $s(n)$ denote the sum of digits in the binary expansion of the integer $n$. Hare, Laishram and Stoll (2011) studied the number of odd integers such that $s(n)=s(n^2)=k$, for a given integer $k\geq 1$. The remaining cases that could not be treated by theses authors were $k\in\{9,10,11,14,15\}$. In this paper we show that there is only a finite number of solutions for $k\in\{9,10,11\}$ and comment on the difficulties to settle the two remaining cases $k\in\{14,15\}$. A related problem is to study the solutions of $s(n^2)=4$ for odd integers. Bennett, Bugeaud and Mignotte (2012) proved that there are only finitely many solutions and conjectured that $n=13,15,47,111$ are the only solutions. In this paper, we give an algorithm to find all solutions with fixed sum of digits value, supporting this conjecture, as well as show related results for $s(n^2)=5$.

math.NT↗

arXiv2021

Products of integers with few nonzero digits

Let $s(n)$ be the number of nonzero bits in the binary digital expansion of the integer $n$. We study, for fixed $k,\ell,m$, the Diophantine system $$ s(ab)=k, \quad s(a)=\ell,\quad \mbox{and }\quad s(b)=m, $$ in odd integer variables $a,b$. When $k=2$ or $k=3$, we establish a bound on $ab$ in terms of $\ell$ and $m$. While such a bound does not exist in the case of $k=4$, we give an upper bound for $\min\{a,b\}$ in terms of $\ell$ and $m$.

math.NT↗

arXiv2021

Maximum order complexity of the sum of digits function in Zeckendorf base and polynomial subsequences

Automatic sequences are not suitable sequences for cryptographic applications since both their subword complexity and their expansion complexity are small, and their correlation measure of order 2 is large. These sequences are highly predictable despite having a large maximum order complexity. However, recent results show that polynomial subsequences of automatic sequences, such as the Thue--Morse sequence, are better candidates for pseudorandom sequences. A natural generalization of automatic sequences are morphic sequences, given by a fixed point of a prolongeable morphism that is not necessarily uniform. In this paper we prove a lower bound for the maximum order complexity of the sum of digits function in Zeckendorf base which is an example of a morphic sequence. We also prove that the polynomial subsequences of this sequence keep large maximum order complexity, such as the Thue--Morse sequence.

math.NT↗

arXiv2020

Discrete correlations of order 2 of generalised Rudin-Shapiro sequences: a combinatorial approach

We introduce a family of block-additive automatic sequences, that are obtained by allocating a weight to each couple of digits, and defining the $n$th term of the sequence as being the total weight of the integer $n$ written in base $k$. Under an additional difference condition on the weight function, these sequences can be interpreted as generalised Rudin-Shapiro sequences, and we prove that they have the same correlations of order 2 as sequences of symbols chosen uniformly and independently at random. The speed of convergence is very fast and is independent of the prime factor decomposition of $k$. This extends recent work of Tahay. The proof relies on direct observations about base-$k$ representations of integers and combinatorial considerations. We also provide extensions of our results to higher-dimensional block-additive sequences.

math.CO↗

arXiv2019

The sum-of-digits function on arithmetic progressions

Let $s_2$ be the sum-of-digits function in base $2$, which returns the number of non-zero binary digits of a nonnegative integer $n$. We study $s_2$ alon g arithmetic subsequences and show that --- up to a shift --- the set of $m$-tuples of integers that appear as an arithmetic subsequence of $s_2$ has full complexity.

math.NT↗

arXiv2018

Somme des chiffres et changement de base

Let $s_a(n)$ denote the sum of digits of an integer $n$ in the base $a$ expansion. Answering, in a extended form, a question of Deshouillers, Habsieger, Laishram, and Landreau, we show that, provided $a$ and $b$ are multiplicatively independent, any positive real number is a limit point of the sequence $\{s_b(n)/s_a(n)\}_{n=1}^{\infty}$. We also provide upper and lower bounds for the counting functions of the corresponding subsequences.

math.NT↗

arXiv2018

Hensel's lemma for general continuous functions

In the present paper, we generalize the well-known Hensel's lifting lemma to any continuous function $f : \mathbb{Z}_p\rightarrow \mathbb{Z}_p$. This answers a question posed by Axelsson and Khrennikov (2016) who showed the validity of Hensel's lemma for $1$- and for $p^α$-Lipschitz functions. For the statement and the proof, we introduce a suitable generalization of the original van der Put series. We use the concept of approximability of continuous functions to give numerical examples.

math.AC↗

arXiv2017

On subwords in the base-$q$ expansion of polynomial and exponential functions

Let $w$ be any word over the alphabet $\{0,1,\ldots, q-1\}$, and denote by $h$ either a polynomial of degree $d\geq 1$ or $h: n\mapsto m^n$ for a fixed $m$. Furthermore, denote by $e_q(w;h(n))$ the number of occurrences of $w$ as a subword in the base-$q$ expansion of $h(n)$. We show that \[ \limsup_{n\to\infty} \frac{e_q(w;h(n))}{\log n}\geq \frac{γ(w)}{l\log q}, \] where $l$ is the length of $w$ and $γ(w)\geq 1$ is a constant depending on a property of circular shifts of $w$. This generalizes work by the second author as well as is related to a generalization of Lagarias of a problem of Erdős.

math.NT↗

arXiv2013

On a conjecture of Dekking : The sum of digits of even numbers

Let $q\geq 2$ and denote by $s_q$ the sum-of-digits function in base $q$. For $j=0,1,...,q-1$ consider $$# \{0 \le n < N : \;\;s_q(2n) \equiv j \pmod q \}.$$ In 1983, F. M. Dekking conjectured that this quantity is greater than $N/q$ and, respectively, less than $N/q$ for infinitely many $N$, thereby claiming an absence of a drift (or Newman) phenomenon. In this paper we prove his conjecture.

math.NT↗

arXiv2011

The sum of digits of polynomial values in arithmetic progressions

Let $q, m\geq 2$ be integers with $(m,q-1)=1$. Denote by $s_q(n)$ the sum of digits of $n$ in the $q$-ary digital expansion. Further let $p(x)\in mathbb{Z}[x]$ be a polynomial of degree $h\geq 3$ with $p(\mathbb{N})\subset \mathbb{N}$. We show that there exist $C=C(q,m,p)>0$ and $N_0=N_0(q,m,p)\geq 1$, such that for all $g\in\mathbb{Z}$ and all $N\geq N_0$, $$#\{0\leq n< N: \quad s_q(p(n))\equiv g \bmod m\}\geq C N^{4/(3h+1)}.$$ This is an improvement over the general lower bound given by Dartyge and Tenenbaum (2006), which is $C N^{2/h!}$.

math.NT↗

arXiv2010

Thue-Morse at Multiples of an Integer

Let (t_n) be the classical Thue-Morse sequence defined by t_n = s_2(n) (mod 2), where s_2 is the sum of the bits in the binary representation of n. It is well known that for any integer k>=1 the frequency of the letter "1" in the subsequence t_0, t_k, t_{2k}, ... is asymptotically 1/2. Here we prove that for any k there is a n<=k+4 such that t_{kn}=1. Moreover, we show that n can be chosen to have Hamming weight <=3. This is best in a twofold sense. First, there are infinitely many k such that t_{kn}=1 implies that n has Hamming weight >=3. Second, we characterize all k where the minimal n equals k, k+1, k+2, k+3, or k+4. Finally, we present some results and conjectures for the generalized problem, where s_2 is replaced by s_b for an arbitrary base b>=2.

math.NT↗

arXiv2010

Extremal orders of the Zeckendorf sum of digits of powers

Denote by s_F(n) the minimal number of Fibonacci numbers needed to write n as a sum of Fibonacci numbers. We obtain the extremal minimal and maximal orders of magnitude of s_F(n^h)/s_F(n) for any h>= 2. We use this to show that for all $>N_0(h) there is an n such that n is the sum of N Fibonacci numbers and n^h is the sum of at most 130 h^2 Fibonacci numbers. Moreover, we give upper and lower bounds on the number of n's with small and large values of s_F(n^h)/s_F(n). This extends a problem of Stolarsky to the Zeckendorf representation of powers, and it is in line with the classical investigation of finding perfect powers among the Fibonacci numbers and their finite sums.

math.NT↗

arXiv2009

A fancy way to obtain the binary digits of $759250125\sqrt{2}$

R. L. Graham and H. O. Pollak observed that the sequence $$u_1=1,\qquad u_{n+1}=\lfloor \sqrt{2} (u_n+1/2)\rfloor, \quad n\geq 1,$$ has the curious property that the sequence of numbers $(u_{2n+1}-2u_{2n-1})_{n\geq 1}$ denotes the binary digits of $\sqrt{2}$. We present an extension of Graham--Pollak's sequence which allows to get -- in a fancy way -- the binary digits of $759250125\sqrt{2}$ and other numbers.

math.NT↗