Algebraic independence of the values of power series with unbounded coefficients

Abstract

Many mathematicians have studied the algebraic independence over Q of the values of gap series, and the values of lacunary series satisfying functional equations of Mahler type. In this paper, we give a new criterion for the algebraic independence over Q of the values ∑∞ n=0 t(n)β −n for distinct sequences (t(n))n=0 of nonnegative integers, where β is a fixed Pisot or Salem number. Our criterion is applicable to certain power series which are not lacunary. Moreover, our criterion does not use functional equations. Consequently, we deduce the algebraic independence of certain values ∑∞ n=0 t1(n)β −n, . . . , ∑∞ n=0 tr(n)β −n satisfying lim n→∞,ti−1(n) ̸=0 ti(n) ti−1(n) = ∞ (i = 2, . . . , r) for any positive real number M . 1 The transcendence of the values of power series with bounded coefficients We introduce notation which we use throughout this paper. Let N (resp. Z) be the set of nonnegative integers (resp. positive integers). For a real number x, we denote the integral and fractional parts of x by ⌊x⌋ and {x}, respectively. We use the Landau symbols o,O, and the Vinogradov symbols ≫,≪ with their regular meanings. For a sequence of integers t = (tn) ∞ n=0, put S(t) := {n ∈ N | tn ̸= 0}. and