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Calculating the density of solutions of equations related to the Pólya–Ostrowski group through Markov chains

Volume 186 / 2018

Dario Spirito Acta Arithmetica 186 (2018), 319-335 MSC: Primary 11B05; Secondary 11A07, 11A63, 13F05, 13F20. DOI: 10.4064/aa170605-6-3 Published online: 9 November 2018


Motivated by a problem in the theory of integer-valued polynomials, we investigate the natural density of the solutions of equations of the form $\theta_uu_q(n)+\theta_ww_q(n)+\theta_2\frac{n(n+1)}{2}+\theta_1n+\theta_0\equiv 0\bmod d$, where $d,q\geq 2$ are fixed integers, $\theta_u,\theta_w,\theta_2,\theta_1,\theta_0$ are parameters and $u_q$ and $w_q$ are functions related to the $q$-adic valuations of the numbers between 1 and $n$. We show that the number of solutions of this equation in $[0,N)$ satisfies a recurrence relation, with which we can associate, for any pair $(d,q)$, a stochastic matrix and a Markov chain. Using this interpretation, we calculate the density for $\theta_u=\theta_2=0$ and for $\theta_u=1$, $\theta_w=\theta_2=\theta_1=0$ and either $d\,|\,q$ or $d$ and $q$ are coprime.


  • Dario SpiritoDipartimento di Matematica e Fisica
    Università degli Studi “Roma Tre”
    Roma, Italy

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