Let {fn} n≥1 be an infinite iterated function system on [0, 1] satisfying the open set condition with the open set (0, 1) and let Λ be its attractor. Then to any x ∈ Λ (except at most countably many points) corresponds a unique sequence {an(x)} n≥1 of integers, called the digit sequence of x, such thatWe investigate the growth speed of the digits in a general infinite iterated function system. More precisely, we determine the dimension of the setfor any infinite subset B ⊂ N, a question posed by Hirst for continued fractions. Also we generalize Łuczak's work on the dimension of the set {x ∈ Λ : an(x) ≥ a b n for infinitely many n ∈ N} with a, b > 1. We will see that the dimension of the sets above is tightly connected with the convergence exponent of the contraction ratios of the sequence {fn} n≥1 .1. Introduction. We first recall the definition of an infinite iterated function system. For a thorough study and foundations of the theory of infinite iterated function systems, one is referred to the works of Hanus, Mauldin, and Urbański [HaMU], Mauldin and Urbański [MU1, MU2] or their monograph [MU3]. Here we follow the notation used in [JorR] by Jordan and Rams.Let {f n } n≥1 be a sequence of functions with(ii) Contraction property: there exists an integer m and a real number 0 < ρ < 1 such that for any (a 1 , . . . , a m ) ∈ N m and x ∈ [0, 1],(iii) Open set condition: for any i = j ∈ N, f i ((0, 1)) ∩ f j ((0, 1)) = ∅.

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Let $\{f_{n}\}_{n\geq 1}$ be an infinite iterated function system on $[0,1]$ and let $\unicode[STIX]{x1D6EC}$ be its attractor. Then, for any $x\in \unicode[STIX]{x1D6EC}$, it corresponds to a sequence of integers $\{a_{n}(x)\}_{n\geq 1}$, called the digit sequence of $x$, in the sense that $$\begin{eqnarray}x=\lim _{n\rightarrow \infty }f_{a_{1}(x)}\circ \cdots \circ f_{a_{n}(x)}(1).\end{eqnarray}$$ In this note, we investigate the size of the points whose digit sequences are strictly increasing and of upper Banach density one, which improves the work of Tong and Wang and Zhang and Cao.

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