Thue-Morse at Multiples of an Integer

  title={Thue-Morse at Multiples of an Integer},
  author={Johannes F. Morgenbesser and Jeffrey Shallit and Thomas Stoll},
  journal={arXiv: Number Theory},

Sums of Digits and the Distribution of Generalized Thue-Morse Sequences

This paper gives an explicit formulation of the exact minimal value of M such that every M consecutive terms in tq,n cover the residue system of n, i.e., {0, 1, . . . , n− 1}.

Transcendence of Digital Expansions Generated by a Generalized Thue-Morse Sequence

The combinatorial transcendence criterion is applied to find that, for a non-periodic generalized Thue-Morse sequence tak∞, no subsequence of the form (a(N +nl)) ∞=0 (where N ≥ 0 and l > 0) is periodic.

Transcendence of digital expansions and continued fractions generated by a cyclic permutation and $k$-adic expansion

In this article, first we generalize the Thue-Morse sequence $(a(n))_{n=0}^\infty$ (the generalized Thue-Morse sequences) by a cyclic permutation and $k$ -adic expansion of natural numbers, and


It is shown that the abelian complexity of vtm, i.e., the number of Parikh vectors of length n, is O (log n) with constant approaching 3 4 (assuming base 2 logarithm), and it is Ω (1) with constants 3 (and these are the best possible bounds).


We study the combinatorics of vtm, a variant of the Thue-Morse word generated by the non-uniform morphism 0 7! 012, 1 7! 02, 2 7! 1 starting with 0. This infinite ternary sequence appears a lot in

Thue-Morse Along Two Polynomial Subsequences

  • T. Stoll
  • Computer Science, Mathematics
  • 2015
The aim of the present article is to give explicit bounds on recent developments on the distribution of symbols in polynomial subsequences of the Thue–Morse sequence by highlighting effective results.

On monochromatic arithmetic progressions in binary words associated with block-counting functions

. Let e v ( n ) denote the number of occurrences of a fixed block v of digits in the binary expansion of n ∈ N . In this paper we study monochromatic arithmetic progressions in the class of binary

The level of distribution of the Thue–Morse sequence

The level of distribution of a complex-valued sequence $b$ measures the quality of distribution of $b$ along sparse arithmetic progressions $nd+a$. We prove that the Thue–Morse sequence has level of

Nondeterministic Automatic Complexity of Almost Square-Free and Strongly Cube-Free Words

It is proved that such complexity A N of a word x of length n satisfies A_N(x) = b(n) − A N (x) and that there is no constant upper bound on D for strongly cube- free words in a ternary alphabet, nor for cube-free Words in a binary alphabet.

Nondeterministic Automatic Complexity of Overlap-Free and Almost Square-Free Words

It is proved that the nondeterministic automatic complexity of a word x of length n is bounded by b(n):=\lfloor n/2\rfloor + 1, which enables to define the complexity deficiency $D(x)=b(n)-A_N(x)$.



Stolarsky's conjecture and the sum of digits of polynomial values

Let s q (n) denote the sum of the digits in the q-ary expansion of an integer n. In 1978, Stolarsky showed that lim n→∞ inf s 2 (n 2 )/s 2 (n) = 0. He conjectured that, just as for n 2 , this limit

On digit sums of multiples of an integer

Somme des chiffres et transcendance

Let/? be a prime number, and let Sp(n) be thé sum ofthe digits ofn in thep-adic expansion. We prove thé following results: Let R be a polynomial in Q[X] with ^(M)c=M. Thé formai séries ^^o5p(R(n))X

On the number of binary digits in a multiple of three

shows a definite preponderance of those containing an even number of one digits over those containing an odd number. Indeed L. Moser has conjectured that this strange behavior persists forever and

A summation formula related to the binary digits

Let s be the smallest integer such that k ≤ b s . Then b s−1 < k. Choose t ∈ {s, s + 1, . . . , s + r − 1} such that (b − 1)t ≡ c (mod r)

    We claim that if 1 ≤ k ≤ b t , then s b (k(b t − 1)) = (b − 1)t. To see this, note that for p, t ≥ 1 and all k with 1 ≤ k < b t we have s b (pb t − k) = s b (p − 1) + (b − 1)t − s b (k − 1)

      Let b, r, k be positive integers with gcd(b − 1, r) = 1, and let a, c be any integers. Then there exists an integer n < b r+1 k 3 such that n ≡ a (mod k) and s b (n) ≡ c