COMBINATORIAL AND ARITHMETICAL PROPERTIES OF LINEAR NUMERATION SYSTEMS PETER J. GRABNER†, PETER KIRSCHENHOFER‡, AND ROBERT F. TICHY * Dedicated to the memory of P´ al Erd˝ os Abstract. We extend a result of J. Alexander and D. Zagier on the Garsia entropy of the Erd˝ os measure. Our investigation heavily relies on methods from combinatorics on words. Furthermore, we introduce a new singular measure related to the Farey tree. 1. Introduction This paper is devoted to the investigation of multiplicities of representations and related combinatorial and probabilistic questions for a special class of linear numeration systems. Non-uniqueness of radix expansions was extensively studied by P´ al Erd˝ os and his coauthors, see for instance [14]. We will study numeration systems given by a linear recurring base sequence G n+m = G n+m−1 + ··· + G n for n ≥ 0 (1.1) G k = G k−1 + ··· + G 0 +1 for 0 ≤ k < m. (1.2) Any positive integer n can be represented in a digital expansion (1.3) n = L  ℓ=0 δ ℓ (n)G ℓ with digits δ ℓ {0, 1} for 0 ≤ ℓ ≤ L, where the digits are computed by the greedy algorithm: there is a unique integer L such that G L ≤ n<G L+1 . Then n can be written as n = δ L G L + n L with 0 ≤ n L <G L and by iterating this procedure with n L the expansion (1.3) is obtained. An extensive description of digital expansions with respect to linear recurring base sequences is given in [15, 19, 20, 21]. In [19], especially dynamical properties of such expansions are investigated. The corresponding shift transformation is the classical β -shift investigated by A. R´ enyi and W. Parry [33, 35]. Recurrence (1.1) has the property that its dominating characteristic root β> 1 satisfying β m = β m−1 + ··· + β +1 m ≥ 2 Date : July 5, 2002. 1991 Mathematics Subject Classification. Primary: 11K55; Secondary: 05A17, 11A63, 28D20. Key words and phrases. digital representations, entropy, generating functions. † This author is supported by the START-project Y96-MAT of the Austrian Science Fund. ‡ This author is supported by the Austrian Science Fund (FWF) grant P14200-MAT. * This author is supported by the Austrian Science Fund (FWF) grant S8307-MAT. 1