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ЧЕБЫШЕВСКИЙ СБОРНИК
Том 24. Выпуск 4.
УДК 517 DOI 10.22405/2226-8383-2023-24-4-341-344
Коэффициент и арифметическая сложность объединения п!
А. Дуаа, М. Мейсами
Дуаа Абдулла — Московский физико-технический институт (Москва). e-mail: duaal992abdullah@gmail.com
Махди Мейсами — Университет Исфахана (г. Исфахан, Иран). e-mail: meisam67@gmail.com
Аннотация
В этой статье мы покажем, что факторная сложность бесконечного слова Fb определяемая путем объединения базовых b представлений п! полна. Затем мы покажем, что арифметическая сложность этого слова также является полной. С другой стороны, Fь это дизъюнктивное слово. В теории чисел такой вид слов называется богатыми цифрами.
Ключевые слова: факторная сложность, равнораспределенная по модулю 1, критерий Вейля, цифровые задачи, факториалы.
Библиография: 5 названий. Для цитирования:
А. Дуаа, М. Мейсами. Коэффициент и арифметическая сложность объединения п! // Чебы-шевский сборник, 2023, т. 24, вып. 4, с. 341-344.
CHEBYSHEVSKII SBORNIK Vol. 24. No. 4.
UDC 517 DOI 10.22405/2226-8383-2023-24-4-341-344
Factor and arithmetic complexity of concatenating the n!
A. Duaa, M. Meisami
Duaa Abdullah — Moscow Institute of Physics and Technology (Moscow, Russia).
e-mail: duaal992abdullah@gmail.com
Mahdi Meisami — University of Isfahan (Isfahan, Iran).
e-mail: meisam67@gmail.com
Abstract
In this paper, we show that factor complexity of the infinite word Fь is defined by-concatenating base-Ь representations of the n! is full. Then we show that the arithmetic complexity of this word is full as well. On the other hand, Fь is a disjunctive word. In number theory, this kind of words is called rich numbers.
1
factorials.
Bibliography: 5 titles.
For citation:
A. Duaa, M. Meisami, 2023, "Factor and arithmetic complexity of concatenating the n!" // Cheby-shevskii sbornik, vol. 24, no. 4, pp. 341-344.
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A. „ Lynn. M. Mci'ica.Mii
1. Introduction and preliminaries
Applying combinatorial analysis to finite or infinite words is the domain of word combinatorics, a field at the intersection of theoretical computer science and discrete mathematics. This area of mathematics evolved from number theory, group theory, probability, and of course combinatorics. The basics of word combinatorics are given for instance in [5].
1.1. Combinatorics on words
A nonempty finite set E is called an alphabet. The elements of the set E are called letters. The alphabet consisting of & symbols from 0 to 6—1 will then be denoted by E^ = {0,..., b—1}. A word w is a sequence of letters. The finite word w can be considered as a function of w : {1, ■ ■ ■ , |w|} ^ E, where w[f] is the letter in the ith position. The length of the word |w| is the number of letters contained in it. The empty word is denoted by e. Then we introduce infinite words as functions w : N ^ E. The set of all finite words over E is denoted by E*, and E+ = E* \ {e}; the set of all infinite words is denoted by EN.
The concatenation of the finite words U = u[1] •••U[n], |u| = n and w = w[1] •••w[m], |w| = m is the word
s = uw = u = u[1] ■ ■ ■ uMw[1] ■ ■ ■ w[m], |s| = |u| + |w| = n + m.
=
called a factor of the word w. The set of all factors of w is denoted by £w and it is called language generated by w. If s = e, then U is called a prefix of the word w, if v = e, it is named a suffix. The factor w[f]w[f + 1] ■ ■ ■ w[j] where i ^ j is denoted by w[f ■ ■ ■ j].
This work investigates concatenating words from a complexity perspective. There are numerous approaches to quantify the complexity of a word over a finite alphabet. The complexity function will be used as our primary measure of complexity. This function was first studied by Hedlund and Morse in 1938 [4].
Definition 1.1. The factor complexity of a finite or infinite word w is the function k ^ pw(k), which, for each integer k, give the number pw(k) of distinct factors of length n in that word.
It is clear that the factor complexity is between zero and (#E)fc. If pw(k) = (#E)fc, then the word w is said to have full factor complexity. This kind of words are called disjunctive word [4].
It is also easy to see that the factor complexity of any infinite word is a non-decreasing function, and the complexity of a finite word first increases, then decreases to zero.
The arithmetic complexity of an infinite word is the function that counts the number of words of a specific length composed of letters in arithmetic progression (and not only consecutive). In fact, it's a generalization of the complexity function. This concept was introduced by Avgustinovich and Frid in [1].
Definition 1.2. Let w = (an)n^o e EN. The arithmetic closure of w is the set
A(w) = {aiai+dai+2d ■ ■ ■ ai+kd | d ^ 1,k ^ 0}.
The arithmetic complexity of w is the function aw mapping n to the number aw(n) of words with length n in A(w).
If aw(fc) = (#E)fc, then the word w is said to have full arithmetic complexity. The following statement immediately follows from the definition:
Proposition 1.3. Let w e EN and #E = k. Then for all n e N we have
1 ^ pw(n) ^ aw(n) ^ kn.
1.2. Ergo die theory
In mathematics, a sequence (sn)n^0 of real numbers is said to be equidistributed or uniformly distributed on a non-degenerate interval [a, b], if the proportion of terms that fall into a sub-interval is proportional to the length of this interval, i.e., if for any sub-interval [c, d] of [a, b] we have
#{{si,...,sn}n [c,d]) _ d — c n b — a
The theory of uniform distribution modulo 1 deals with the distribution behavior of sequences of real numbers.
De pinition 1.4. A sequence (an)n^o of real numbers is said to be equidistributed modulo 1 or uniformly distributed modulo 1 if the sequence of fractional parts of (an)n^0 is equidistributed in the interval [0,1].
Theorem 1.5. (Wevl's criterionf2]) A sequence (an)n^0 is equidistributed modulo 1 if and only if for all non-zero integers N,
1 n
lim -V e2wiNaJ _ 0. n^-^ n —' 3=1
Lemma 1.6. [3] The fractional part of the sequence (log(n!))ra^0 is dense in [0,1].
Theorem 1.7. [3] If k is any positive integer having m digits, there exists a positive integer n such that the first m digits of n! constitute the integer k.
Also we can state this result for any arbitrary base. Let dm ... dn be a word over S^ with dm _ 0. There exists n such that the base^ expansion of n! starts with dm ... dn.
2. Main result
2.1. Statement of problem
The infinite word F _ Fb :_ (bfn)n^0 is defined by concatenating non-negative base-6 ^ 2 representation of the recursive n!.
by concatenating base-10 of the recursive n!:
F :_ (fn)n>o _ 1 1 2 6 24 120 720 5040 40320 ■ ■ ■ .
What is the factor complexity of the F, i-e■ p$(k) ? What about arithmetic complexity, i.e., &$(k) ? In fact, this problem can be easily generalized for any natural bases. Theorem 2.1. (i) The factor complexity of the infinite word Fb is full, (ii) The arithmetic complexity of the infinite word Fb is full.
Proof, (i) Let the alphabet for base-Ms _ {0, ■ ■ ■ ,b — 1}. Then Fb g Now we want to find
k^ #{U■■■ fi+k-i\i > 0}.
By Lemma 1.2 and Theorem 1.2, we claim that (bfn)n^0 is equidistributed modulo 1. Then we have a same result such Theorem 1.2, but for an arbitrary bases, i.e., there exists an n such that the ^expansion of n! begins with these digits. On the other hand, each word t g S+ will appearance at least one position in F&, i-e-, Fb[i ■ ■ ■ j] _ t, because there exist s g S+ such that it begins by t. Hence, CFb _ S+, and Fb is full factor complexity, i.e.,
PFb (fc) = (#(Xb))fc = bk.
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А. Дуаа, М. Мейсами
(ii) In the previous part we show that p^(k) = bk. Now bv Proposition 1.1, we can say that
for all к e N
bk < (k) < bk.
This inequality is true for all natural numbers k, this implies that agb (k) = bk.
СПИСОК ЦИТИРОВАННОЙ ЛИТЕРАТУРЫ
1. S.V. Avgustinovich, D.G. Fon-Der-Flaass, and A.E. Frid, Arithmetical complexity of infinite words, Languages and Combinatorics III (Proc. 3rd ICWLC, Kyoto, March 2000), World Scientific, Singapore (2003), 51-62.
2. L.Kuipers, H. Niderreiter, Uniform Distribution of Sequences, Pure and applied mathematics,A Wiley-Inter science publication, New York (1974).
3. J. E. Maxfield, A Note on nl, Mathematics Magazine, 43 (1997), 64-67.
4. M. Morse, G. A. Hedlund. Symbolic dynamics. Amer. J. Math., 60, (1938): 815-866.
5. M. Rigo, Formal languages, automata and numeration system,s 1: Introduction to combinatorics on words, John WTilev k, Sons, (2014).
REFERENCES
1. S.V. Avgustinovich, D.G. Fon-Der-Flaass, and A.E. Frid. 2003, "Arithmetical complexity of infinite words", Languages and Combinatorics III (Proc. 3rd ICWLC, Kyoto, March 2000), World Scientific, Singapore, pp. 51-62.
2. L. Kuipers and H. Niderreiter. 1974, "Uniform Distribution of Sequences", Pure and applied mathematics, A Wiley-Interscience publication, New York.
3. J. E. Maxfield. 1997, "A Note on n\, Mathematics Magazine, 43 (1997), pp. 64-67.
4. M. Morse and G. A. Hedlund. 1938, "Symbolic dynamics", Amer. J. Math., 60, pp. 815—866.
5. M. Rigo. 2014, "Formal languages, automata and numeration systems 1: Introduction to combinatorics on words", John Wiley & Sons.
Получено: 27.09.2023 Принято в печать: 11.12.2023
