Modular automata

• Thomas N. Hibbard ^[1] ; Camilo A. Jadur ^[1] ; Jorge F. Yazlle ^[1]
  1. [1] Universidad Nacional de Salta

     Universidad Nacional de Salta

     Argentina

     
• Localización: Revista de la Unión Matemática Argentina, ISSN 0041-6932, ISSN-e 1669-9637, Vol. 67, Nº. 1, 2024, págs. 229-244
• Idioma: inglés
• Enlaces
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• Resumen

  • Let M and b be integers greater than 1, and let p be a positive probability vector for the alphabet Ab={0,…,b−1}. Let us consider a random sequence w0,w1,…,wj over AbAb, where the wi's are independent and identically distributed according to p. Such a sequence represents, in base b, the number n=∑ji=0wibj−i. In this paper, we explore the asymptotic distribution of n mod M, the remainder of nn divided by M. In particular, by using the theory of Markov chains, we show that if M and b are coprime, then n mod M exhibits an asymptotic discrete uniform distribution, independent of p; on the other hand, when Mand b are not coprime, n mod M does not necessarily have a uniform distribution, and we obtain an explicit expression for this limiting distribution.