ALMOST EVERY NUMBER HAS A CONTINUUM. INTRODUCTION. Let β belong to (1, 2) and write Σ =

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1 ALMOST EVERY NUMBER HAS A CONTINUUM OF β-expansions NIKITA SIDOROV 1. INTRODUCTION. Let β belong to (1, 2) and write Σ = 1 {0, 1}. For each x 0 we will call any representation of x of the form (1) x = π β (ε 1, ε 2,... ) = ε n β n, with (ε 1, ε 2,... ) in Σ, a β-expansion of x. It is obvious that, since the ε n are 0s or 1s, x must belong to the interval I β := [0, 1/(β 1)]. At the same time, every x in I β has at least one β-expansion, namely, the greedy β-expansion. It can be obtained as follows: if x lies in [0, 1), then we use the standard greedy algorithm (the same as in the decimal or binary number system): n=1 ε 1 = βx, ε n := β {β{... {βx}... }} (n = 2, 3,... ), }{{} n 1 curly braces where denotes the integral part of a number and { } stands for its fractional part. If x belongs to [1, 1/(β 1)), then we put l = min {k 1 : x β 1 β k (0, 1)} and apply the greedy algorithm to x β 1 β l to obtain the digits ε l+1, ε l+2, etc. Finally, if x = 1/(β 1), then ε n = 1 for all n. The question then arises: Are there any other β-expansions of a given x besides the greedy one; if yes, how many? In [2, Theorem 2] P. Glendinning and the author proved in particular that, if β is greater than a certain constant, then there exists a continuum of points x for which the greedy β-expansion is the only β-expansion. The aim of this note is to present a result in the opposite direction. Note that in [5, Appendix A] A. Vershik and the author have given the full description of β-expansions for any x in the case β = (1 + 5)/2. The result we wish to prove in the present note reads as follows: Theorem 1. Almost every point x of I β has a continuum of different β-expansions. 1

2 2 NIKITA SIDOROV Note that P. Erdös, I. Joó, and V. Komornik have shown in [1, Theorem 3] that if β < (1+ 5)/2, then in fact every x in (0, 1/(β 1)) has this property. Thus, we can confine ourselves to the case β (1 + 5)/2. 2. AUXILIARY LEMMA. We begin with an auxiliary claim that is in fact known; for example, [2, Theorem 2] yields it in even a stronger form. In order to keep our proof self-contained and elementary, we furnish its proof as well. Let m β denote normalized Lebesgue measure on I β. Lemma 2. For any β (1+ 5)/2 there exists a set E β with m β (E β ) = 1 such that each x in E β has at least two different β-expansions. Proof. Since β belongs to [(1 + 5)/2, 2), there exists m = m(β) 2 such that (2) 1 + β m+1 < 1 β 1 ; specifically, we can take m = β 1 log β β (for β = (1 + 5)/2 we have β 1 = β 1, 2 β = β 2, whence β 1 log = 1). β 2 β First consider x in (0, 1), and assume that its greedy expansion is of the form (ε 1,..., ε n, 1, 0,..., 0, ε }{{} n+m+1,... ). m 1 We can construct a different β-expansion for x. Namely, if x = n ε jβ j, then x x = β n 1 + j=n+m+1 ε j β j [β n 1, β n 1 + β n m ], because n+m+1 ε jβ j β n m (the property of a greedy expansion). On the other hand, we infer from (2) that whence β n 1 + β n m < β n 2 + β n 3 + = β n 1 β 1, x x < β n 2 + β n 3 +

3 CONTINUUM OF β-expansions 3 as well. This means that if we put ε n+1 = 0, it is possible to find (ε n+2, ε n+3,... ) in Σ such that x = ε jβ j. By our construction, ε n+1 ε n+1. For each m 1 and each vector (i 1,..., i m ) in {0, 1} m we will call the set of all sequences ε with ε 1 = i 1,..., ε m = i m, the cylinder [i 1... i m ]. It now suffices to show that for almost every x in (0, 1) some shift of its greedy expansion belongs to the cylinder [10 m 1 ]. This can be done using Poincaré s famous recurrence theorem (see [6, Theorem 1.4]). Theorem 3. (H. Poincaré, 1890). Let (X, A, µ) be a measure space with µ(x) <. If T : X X is a map such that µ(t 1 E) = µ(e) for every E in A (such a map is called measure-preserving) and if F in A has a positive µ-measure, then for µ-almost every point x of X there exists n in N such that T n x belongs to F. Let X β denote the space of 0-1 sequences that represent greedy expansions for the numbers from [0, 1). The map π β : X β [0, 1) given by (1) is known to be one-to-one except for a certain countable set, a situation analogous to the decimal expansions (see [3]). Recall that two measures µ 1 and µ 2 are called equivalent if µ 1 (E) = 0 for some set E if and only if µ 2 (E) = 0. Theorem 2 in [4] asserts that there exists a measure ν β on X β that is equivalent to the π β -preimage of Lebesgue measure and is preserved by the one-sided shift τ β on X β : τ β (ε 1, ε 2, ε 3,... ) = (ε 2, ε 3,... ). Moreover, from the same theorem it follows that ν β is positive on any cylinder set in X β, in particular, on [10 m 1 ]. Poincaré s recurrence theorem applied to our case implies that for ν β -almost every sequence (ε 1, ε 2,... ) in X β there exists n such that the nth iterate of τ β of (ε 1, ε 2,... ) is a sequence of the form (10 m 1... ). Hence for almost every greedy expansion some power of its shift belongs to the cylinder [10 m 1 ]. Finally, if x lies in [1, 1/(β 1)), it must belong to [ l 1 0 β j, l 0 β j) for some l 1. We put y = x l 1 β j, a point of (0, 1), and see that generic y leads to generic x, because Lebesgue measure is translationinvariant. 3. PROOF OF THEOREM 1. It suffices to show that there exists a subset F β of I β of full Lebesgue measure with the following property: for any x in F β, for an arbitrary β-expansion (ε 1, ε 2,... ) of x, and for every k 1, there exists n 1 such that x admits a β-expansion (ε 1,..., ε k+n, ε k+n+1, ε k+n+2,... ) with ε k+n+1 ε k+n+1. We will call this property of a β-expansion Property B ( B for branching ). Our argument is of the type single branching implies infinite branching.

4 4 NIKITA SIDOROV Let Σ k = k 1 {0, 1} and for each (ε 1,..., ε k ) in Σ k define ε1...ε k by [ k ] k ε1...ε k = ε j β j, ε j β j + β k I β ; β 1 i.e., x is in ε1...ε k if and only if one of its β-expansions begins with (ε 1,..., ε k ). Obviously, I β = ε1...ε k, (ε 1,...,ε k ) Σ k but this union is not disjoint (which is why there are multiple β- expansions!). We make it disjoint in the standard way, namely, by taking all necessary intersections and set-theoretic differences. We thus obtain, for any fixed k = 1, 2,..., the disjoint union where I β = αk = r k q=1 p i=1 αk, ε (i) 1...ε(i) k for some p = p(α, k). Then x is an element of αk if and only if an arbitrary β-expansion of x begins with (ε (i) 1,..., ε (i) k ) for some i = 1, 2,..., p. Put { } k F αk = x αk : β k x ε (i) j βk j E β, 1 i p, where E β is the set of full measure defined in Lemma 2. In other words, for an x to be in F αk means that for an arbitrary β-expansion (ε 1, ε 2,... ) of x we can find n 1 such that there is a β-expansion of x differing from (ε 1, ε 2,... ) in the (n + k)th coordinate. All that is left is to prove genericity and to perform all the necessary unions and intersections. First, we show that F αk has full measure for any fixed pair (α, k). Let S (i) be the following affine map from αk onto I β : where a (i) = k ε(i) j S (i) (x) = β k (x a (i) ), β j. Then p ( F αk = ) S (i) 1 (Eβ ). i=1

5 CONTINUUM OF β-expansions 5 Since each S (i) is an affine bijection, we conclude on the basis of Lemma 2 that m β (F αk ) = m β ( αk ). This is so because, if I 1 and I 2 are two intervals, m 1 and m 2 are the corresponding normalized Lebesgue measures, and f : I 1 I 2 is an affine map, then clearly m 2 (f(e)) = m 1 (E) for any Borel subset E of I 1. Define r k F β = F αk. k=1 α=1 Since F β is an intersection of sets of full measure, m β (F β ) = 1. Consider a point x of F β. Fix a natural number k and a certain β-expansion of x of the form (1). Then x belongs to r k α=1 F αk, and by our construction, there exists a unique α in [1, r k ] N such that x lies in F αk and (ε 1,..., ε k,... ) has Property B. The proof is complete. Remark 4. We believe that using a more detailed (ergodic) analysis of return times n = n(k) (see the definition of Property B), it is possible to describe the continuum of β-expansions for a generic x in a more quantitative way. This will be attempted elsewhere. ACKNOWLEDGMENT. The author was supported by the EPSRC grant no GR/R61451/01. References [1] P. Erdös, I. Joó, and V. Komornik, Characterization of the unique expansions 1 = i=1 q ni and related problems, Bull. Soc. Math. Fr. 118 (1990) [2] P. Glendinning and N. Sidorov, Unique representations of real numbers in noninteger bases, Math. Res. Lett. 8 (2001) [3] W. Parry, On the β-expansions of real numbers, Acta Math. Acad. Sci. Hung. 11 (1960) [4] A. Rényi, Representations for real numbers and their ergodic properties, Acta Math. Acad. Sci. Hung. 8 (1957) [5] N. Sidorov and A. Vershik, Ergodic properties of Erdös measure, the entropy of the goldenshift, and related problems, Monatsh. Math. 126 (1998) [6] P. Walters, Introduction to Ergodic Theory, Springer-Verlag, New York, Department of Mathematics, UMIST, P.O. Box 88, Manchester M60 1QD, United Kingdom. Nikita.A.Sidorov@umist.ac.uk