On square-free arithmetic progressions in infinite words

The palindromic length $\text{PL}(v)$ of a finite word $v$ is the minimal number of palindromes whose concatenation is equal to $v$. In 2013, Frid, Puzynina, and Zamboni conjectured that: If $w$ is an infinite word and $k$ is an integer such that $\text{PL}(u)\leq k$ for every factor $u$ of $w$ then $w$ is ultimately periodic. Suppose that $w$ is an infinite word and $k$ is an integer such $\text{PL}(u)\leq k$ for every factor $u$ of $w$. Let $\Omega(w,k)$ be the set of all factors $u$ of $w$ that have more than $\sqrt[k]{k^{-1}\vert u\vert}$ palindromic prefixes. We show that $\Omega(w,k)$ is an infinite set and we show that for each positive integer $j$ there are palindromes $a,b$ and a word $u\in \Omega(w,k)$ such that $(ab)^j$ is a factor of $u$ and $b$ is nonempty. Note that $(ab)^j$ is a periodic word and $(ab)^ia$ is a palindrome for each $i\leq j$. These results justify the following question: What is the palindromic length of a concatenation of a suffix of $b$ and a periodic word $(ab)^j$ with "many" periodic palindromes? It is known that $\lvert\text{PL}(uv)-\text{PL}(u)\rvert\leq \text{PL}(v)$, where $u$ and $v$ are nonempty words. The main result of our article shows that if $a,b$ are palindromes, $b$ is nonempty, $u$ is a nonempty suffix of $b$, $\vert ab\vert$ is the minimal period of $aba$, and $j$ is a positive integer with $j\geq3\text{PL}(u)$ then $\text{PL}(u(ab)^j)-\text{PL}(u)\geq 0$.

The Monadic Theory of Morphic Infinite Words and Generalizations

The article surveys some decidability results for DPDAs on infinite words\n(omega-DPDA). We summarize some recent results on the decidability of the\nregularity and the equivalence problem for the class of weak omega-DPDAs.\nFurthermore, we present some new results on the parity index problem for\nomega-DPDAs. For the specification of a parity condition, the states of the\nomega-DPDA are assigned priorities (natural numbers), and a run is accepting if\nthe highest priority that appears infinitely often during a run is even. The\nbasic simplification question asks whether one can determine the minimal number\nof priorities that are needed to accept the language of a given omega-DPDA. We\nprovide some decidability results on variations of this question for some\nclasses of omega-DPDAs.\n

In 2007, Grytczuk conjectured that for any sequence ( ℓ i ) i ≥ 1 (\ell _i)_{i\ge 1} of alphabets of size 3 3 there exists a square-free infinite word w w such that for all i i , the i i -th letter of w w belongs to ℓ i \ell _i . The result of Thue from 1906 implies that there is an infinite square-free word if all the ℓ i \ell _i are identical. On the other hand, Grytczuk, Przybyło and Zhu showed in 2011 that it also holds if the ℓ i \ell _i are of size 4 4 instead of 3 3 . In this article, we first show that if the lists are of size 4 4 , the number of square-free words is at least 2.45 n 2.45^n (the previous similar bound was 2 n 2^n ). We then show our main result: we can construct such a square-free word if the lists are subsets of size 3 3 of the same alphabet of size 4 4 . Our proof also implies that there are at least 1.25 n 1.25^n square-free words of length n n for any such list assignment. This proof relies on the existence of a set of coefficients verified with a computer. We suspect that the full conjecture could be resolved by this method with a much more powerful computer (but we might need to wait a few decades for such a computer to be available).

Consider all the arithmetic progressions of odd numbers, no term of which is of the form 2 k + p, where k is a positive integer and p is an odd prime. Erdős ever asked whether all these progressions can be obtained from covering congruences. In this paper, we characterize all arithmetic progressions in which there are positive proportion natural numbers that can be expressed in the form 2 k +p, and give a quantitative form of Romanoff’s theorem on arithmetic progressions. As a corollary, we prove that the answer to the above Erdős problem is affirmative.

Si q≥2 est un nombre entier, on designe par Sq(n) la somme des chiffres en base q du nombre entier naturel n et par vq(n) sa valuation q-adique. L’objectif de cet article est d’etudier des sommes d’exponentielles de la forme ∑n≤xexp(2iπ(lmSq(n)+km′Sq(n+1)+θn)) afin d’en deduire certaines proprietes statistiques des nombres entiers n pour lesquels Sq(n) et Sq(n+1) appartiennent a des progressions arithmetiques donnees. Ceci permet d’etendre les resultats obtenus par Gelfond en 1968 et ceux obtenus par Mauduit-Sarkozy en 1996.

This is the first part of a two-part paper. Our paper was motivated by two classical papers: A paper of Sir Charles Wheatstone published in 1844 on representing certain powers of an integer as sums of arithmetic progressions and a paper of J. J. Sylvester published in 1882 for determining the number of ways a positive integer can be represented as the sum of a sequence of consecutive integers. There have been many attempts to extend Sylvester Theorem to the number of representations for an integer as the sums of different types of sequences, including sums of certain arithmetic progressions. In this part of the paper, we will make yet one more extension: We will describe a procedure for computing the number of ways a positive integer can be represented as the sums of all possible arithmetic progressions, together with an example to illustrate how this procedure can be carried out. In the process of doing this, we will also give an extension of Wheatstone’s work. In the second part of the paper, we will continue on the problems initiated by Wheatstone by studying certain relationships among the representations for different powers of an integer as sums of arithmetic progressions.

In this paper we answer two recent questions from Charlier et al. (2014) and Harju (2013) about self-shuffling words. An infinite word $w$ is called self-shuffling, if $w=\prod_{i=0}^\infty U_iV_i=\prod_{i=0}^\infty U_i=\prod_{i=0}^\infty V_i$ for some finite words $U_i$, $V_i$. Harju recently asked whether square-free self-shuffling words exist. We answer this question affirmatively. Besides that, we build an infinite word such that no word in its shift orbit closure is self-shuffling, answering positively a question of E. Charlier et al.

Given an infinite word with values in a finite alphabet that admit frequencies for all its factors, the smallest frequency function associates with a given positive integer the smallest frequency for factors of this length, and the recurrence function measures the gaps between successive occurrences of factors. We develop in this paper the relations between the growth orders of the smallest frequency and of the recurrence functions. As proved by M. Boshernitzan, linearly recurrent words are characterized in terms of the smallest frequency function. In this paper, we see how to extend this result beyond the case of linearly recurrent words. We first establish a general relation between the smallest frequency and the recurrence functions. Given a lower bound for the smallest frequency function, this relation provides an upper bound for the recurrence function which involves the primitive of this lower bound. We then see how to improve this relation in the case of Sturmian words where the product of the smallest frequency and the recurrence function is bounded for most of the lengths of factors. We also indicate how to construct Sturmian words having a prescribed behaviour for the smallest frequency function, and for the product of the smallest frequency and the recurrence function.

Infinite words and biprefix codes

Let r and s be multiplicatively independent positive integers. We establish that the r-ary expansion and the s-ary expansion of an irrational real number, viewed as infinite words on {0,1,...,r-1} and {0,1,...,s-1}, respectively, cannot have simultaneously a low block complexity. In particular, they cannot be both Sturmian words.

Introduction 1. Primes in arithmetic progressions 2. The main theorems II.Oscillation theorems for sifting functions 3. The statements 4. Proof of Theorem B2 5. Proof of Proposition 3.1 6. Proof of Proposition 3.2 7. Proof of Theorem C: preliminaries 8. Proof of Theorem C: completion 9. Proof of Theorem Bl 10.Proof of Theorem B3 III.Oscillation theorems for primes 11.Proof of Theorem Al 12. Proof of Theorem A2 13.Proofs of Theorem A3 and Proposition 2.1 I. INTRODUCTION 1. PRIMES IN ARITHMETIC PROGRESSIONSFor x ~ 2 real, q a positive integer, and a an integer coprime to q, let 0 and ~ be defined by (1.1) O(x; q, a) = E logp = qJZq) (1 +~(x; q, a)), p~x p=amodq

We introduce a new, elementary method for studying random differences in arithmetic progressions and convergence phenomena along random sequences of integers. We apply our method to obtain significant improvements on two results. The first improvement is the following. Let l be a positive integer and {u 1 ≥ u 2 ≥ · · · } be a decreasing sequence of probabilities satisfying u n · n 1/(l+1)→∞. Let R = R ω be the random sequence obtained by selecting the natural number n with probability u n . Then every set A of natural numbers with positive upper density contains an arithmetic progression a, a+r, a+2r,..., a+lr of length l + 1 with difference r ∈ R ω . The best previous result (by M. Christ and us) was the condition u n ·n 2−l+1 → ∞ with a logarithmic rate. The new bound is better when l ≥ 4. Our second improvement concerns almost everywhere convergence of double ergodic averages. We construct a (random) sequence {r 1 < r 2 < · · · } of positive integers such that r n /n 2−ϵ → ∞ for all ϵ > 0 and, for any measure preserving transformation T of a probability space, the averages $$\frac{1}{N}\sum\limits_{n < N} {{T^n}{F_1}\left( x \right)} {T^{{r_n}}}{F_2}\left( x \right)$$ converge for almost every x. Our best previous result was the growth rate r n /n (1+1/14)−ϵ → ∞ of the sequence {r n }.

In their valuable book Board games around the world . Bell and Cornelius discuss possible investigations arising from the rather complicated medieval game of Rithmomachia . One such investigation gives rise to the following problem: Find all sets of four positive integers in which three of the four are in arithmetic progression, another three are in geometric progression, and another three are in harmonic progression (that is to say, their reciprocals are in arithmetic progression). The authors give two such sets, and state without proof that they are the only ones, apart from multiplication throughout by a common factor. The proof that this is so is an interesting investigation in itself, and opens up some further mathematics.

Périodes locales et propagation de périodes dans un mot