Redundancy of minimal weight expansions in Pisot bases
Let \(\beta\) be a Pisot number, i.e., an algebraic integer \(\beta>1\) such that all conjugates have modulus less than one. Let \(U=(U_j)_{j\geq 0}\) be a strictly increasing sequence of integers with \(U_0=1\) eventually satisfying a linear recurrence whose characteristic polynomial is equal to the minimal polynomial of \(\beta\). The authors study digital expansions to the base of \(U\) with integer digits, i.e., words \(z=z_kz_{k-1}\ldots z_0\) over the alphabet \({\mathbb Z}\) and their values \(\sum_j z_jU_j\). The weight of a word is defined to be the sum of the absolute values of the digits. Let \(L_U\) be the set consisting of those words that minimize the weight over all expansions of the same value. The authors prove that \(L_U\) is recognized by a finite automaton. They give an asymptotic expression for the summatory function of the number of representations of \(n\) with minimal weight. It consists of a fluctuating main term and an error term. Finally, they show that there is a regular language \(G_U\subseteq L_U\) such that every integer is uniquely representable by an element of \(G_U\). This generalizes \textit{C. Frougny} and \textit{W. Steiner} [J. Math. Cryptol. 2, No. 4, 365--392 (2008; Zbl 1170.11003)] and \textit{P. Grabner} and \textit{C. Heuberger} [``On the number of optimal base 2 representations of integers'', Des. Codes Cryptogr. 40, 25--39 (2006)].
scientific article; zbMATH DE number 5995823
scientific article