aa r X i v : . [ m a t h . N T ] D ec A TWO-DIMENSIONAL UNIVOQUE SET
MARTIJN DE VRIES AND VILMOS KOMORNIK
Abstract.
Let J ⊂ R be the set of couples ( x, q ) with q > x hasat least one representation of the form x = P ∞ i =1 c i q − i with integer coefficients c i satisfying 0 ≤ c i < q , i ≥
1. In this case we say that ( c i ) = c c . . . is anexpansion of x in base q . Let U be the set of couples ( x, q ) ∈ J such that x has exactly one expansion in base q .In this paper we deduce some topological and combinatorial properties ofthe set U . We characterize the closure of U , and we determine its Hausdorffdimension. For ( x, q ) ∈ J , we also prove new properties of the lexicographicallylargest expansion of x in base q . Introduction
Let J be the set consisting of all elements ( x, q ) ∈ R × (1 , ∞ ) such that thereexists at least one sequence ( c i ) = c c . . . of integers satisfying 0 ≤ c i < q for all i ,and(1.1) x = c q + c q + · · · . If (1.1) holds, we say that ( c i ) is an expansion of x in base q , and if the base q isunderstood from the context, we sometimes simply say that ( c i ) is an expansion of x . The numbers c i of an expansion ( c i ) are usually referred to as digits . We denoteby ⌈ q ⌉ the smallest integer larger than or equal to q . The alphabet A q is the set of“admissible” digits in base q , i.e., A q = { , . . . , ⌈ q ⌉ − } .If q > ≤ x ≤ ( ⌈ q ⌉ − / ( q − x inbase q , the so-called quasi-greedy expansion ( a i ( x, q )), may be defined recursivelyas follows. For x = 0 we set ( a i ( x, q )) := 0 ∞ . If x > a i ( x, q ) has already beendefined for 1 ≤ i < n (no condition if n = 1), then a n ( x, q ) is the largest elementof A q satisfying a ( x, q ) q + · · · + a n ( x, q ) q n < x. One easily verifies that ( a i ( x, q )) is indeed an expansion of x in base q . Therefore( x, q ) ∈ J ⇐⇒ q > x ∈ J q := (cid:20) , ⌈ q ⌉ − q − (cid:21) . Let us denote by U the set of couples ( x, q ) ∈ J such that x has exactly one expansion in base q . For example, (0 , q ) ∈ U for every q >
1, but U has muchmore elements. The main purpose of this paper is to describe the topological andcombinatorial nature of U . We will prove the following theorem: Theorem 1.1. (i)
The set U is not closed. Its closure U is a Cantor set . Date : October 13, 2018.2000
Mathematics Subject Classification.
Primary:11A63, Secondary:11B83.
Key words and phrases.
Greedy expansion, beta-expansion, univoque sequence, univoque set,Cantor set, Hausdorff dimension. We recall that a Cantor set is a nonempty closed set having neither interior nor isolatedpoints. (ii)
Both U and U are two-dimensional Lebesgue null sets. (iii) Both U and U have Hausdorff dimension two. As far as we know this two-dimensional univoque set has not yet been inves-tigated. There exists, however, a number of papers devoted to the study of itsone-dimensional sections U := { q > , q ) ∈ U } and U q := { x ∈ J q : ( x, q ) ∈ U } , q > . The study of U started with the paper of Erd˝os, Horv´ath and Jo´o [6] and wasstudied subsequently in [4], [5], [7], [8], [13], [14], [15]. We recall in particular that U and its closure U have Lebesgue measure zero and Hausdorff dimension one.The sets U q have been investigated in [3], [4], [5], [10], [11], [12]. It is known that U q is closed if and only if q does not belong to the null set U , and that its closure U q has Lebesgue measure zero for all non-integer bases q >
1. Moreover, the setof numbers x ∈ J q having continuum many expansions in base q has full Lebesguemeasure for each non-integer q > U byusing the quasi-greedy expansions ( a i ( x, q )). We write for brevity α i ( q ) := a i (1 , q ), i ∈ N := { , , . . . } , q >
1. Note that α ( q ) = ⌈ q ⌉ −
1, the largest admissibledigit in base q . In the statement of the following theorem we use the lexicographicorder between sequences and we define the conjugate of the number a i ( x, q ) by a i ( x, q ) := α ( q ) − a i ( x, q ). If q > c i ∈ A q , i ≥
1, we shall also write c . . . c n instead of c . . . c n and c c . . . instead of c c . . . . Theorem 1.2.
A point ( x, q ) ∈ J belongs to U if and only if a n +1 ( x, q ) a n +2 ( x, q ) . . . ≤ α ( q ) α ( q ) . . . whenever a n ( x, q ) > . Along with the quasi-greedy expansion, we also need the notion of the greedyexpansion ( b i ( x, q )) for x ∈ J q , introduced by R´enyi [17]. It can be defined by aslight modification of the above recursion: if b i ( x, q ) has already been defined forall 1 ≤ i < n (no condition if n = 1), then b n ( x, q ) is the largest element of A q satisfying b ( x, q ) q + · · · + b n ( x, q ) q n ≤ x. Note that the greedy expansion ( b i ( x, q )) of a number x ∈ J q is the lexicographicallylargest expansion of x in base q . We denote the greedy expansion of 1 in base q by( β i ( q )) := ( b i (1 , q )).The rest of this paper is organized as follows. In the next section we give a shortoverview of some basic results on greedy and quasi-greedy expansions, and we provesome new results concerning the coordinate-wise convergence of sequences of theseexpansions. We shall prove (see Theorem 2.7) that the set of numbers x ∈ J q forwhich the greedy expansion of x in base q is not the greedy expansion of a numberbelonging to J p in any smaller base p ∈ (1 , q ) is of full Lebesgue measure andits complement in J q is a set of first category and Hausdorff dimension one. Weshall also prove (see Theorem 2.8) that for each word v := b ℓ +1 ( x, q ) . . . b ℓ + m ( x, q )( ℓ ≥ , m ≥ , x ∈ [0 , Y v ⊂ J q of first category and Hausdorffdimension less than one, such that the word v occurs in the greedy expansion inbase q of every number belonging to J q \ Y v . Using (some of) the results of Section 2we prove Theorem 1.2 in Section 3 and Theorem 1.1 in Section 4. TWO-DIMENSIONAL UNIVOQUE SET 3 Greedy and quasi-greedy expansions
In this paper we consider only one-sided sequences of nonnegative integers. Weequip this set of sequences { , , . . . } N with the topology of coordinate-wise con-vergence. We say that an expansion is infinite if it has infinitely many nonzeroelements; otherwise it is called finite . Using this terminology, the quasi-greedy ex-pansion ( a i ( x, q )) of a number x ∈ J q \ { } is the lexicographically largest infinite expansion of x in base q .The family of all quasi-greedy expansions is characterized by the following propo-sitions (see [1] or [5] for a proof): Proposition 2.1.
The map q ( α i ( q )) is a strictly increasing bijection from theopen interval (1 , ∞ ) onto the set of all infinite sequences ( α i ) satisfying α k +1 α k +2 . . . ≤ α α . . . for all k ≥ . Proposition 2.2.
For each q > , the map x ( a i ( x, q )) is a strictly increasingbijection from J q \ { } onto the set of all infinite sequences ( a i ) satisfying a n ∈ A q for all n ≥ and a n +1 a n +2 . . . ≤ α ( q ) α ( q ) . . . whenever a n < α ( q ) . The quasi-greedy expansions have a lower semicontinuity property for the ordertopology induced by the lexicographic order. More precisely, we have the followingresult.
Lemma 2.3.
Let ( x, q ) ∈ J . Then (i) for each positive integer m there exists a neighborhood W ⊂ R of ( x, q ) such that (2.1) a ( y, r ) . . . a m ( y, r ) ≥ a ( x, q ) . . . a m ( x, q ) for all ( y, r ) ∈ W ∩ J ;(ii) if ( y n , r n ) converges to ( x, q ) in J from below, then ( a i ( y n , r n )) convergesto ( a i ( x, q )) .Proof. (i) We may assume that x = 0. By definition of the quasi-greedy expansionwe have n X i =1 a i ( x, q ) q i < x for all n = 1 , , . . . . For any fixed positive integer m , if ( y, r ) ∈ J is sufficiently close to ( x, q ), then r > ⌈ q ⌉ −
1, i.e., A q ⊂ A r , and n X i =1 a i ( x, q ) r i < y, n = 1 , . . . , m. These inequalities imply (2.1).(ii) If y n ≤ x and r n ≤ q , we deduce from the definition of the quasi-greedyexpansion that ( a i ( x, q )) ≥ ( a i ( y n , r n ))for every n . Equivalently, we have a ( x, q ) . . . a m ( x, q ) ≥ a ( y n , r n ) . . . a m ( y n , r n )for all positive integers m and n . It remains to notice that by the previous part theconverse inequality also holds for each fixed m if n is large enough. (cid:3) The family of greedy expansions has already been characterized by Parry [16]:
MARTIJN DE VRIES AND VILMOS KOMORNIK
Proposition 2.4.
For a given base q > , the map x ( b i ( x, q )) is a strictlyincreasing bijection from J q onto the set of all sequences ( b i ) satisfying b n ∈ A q for all n ≥ and b n +1 b n +2 . . . < α ( q ) α ( q ) . . . whenever b n < α ( q ) . The greedy expansions have the following upper semicontinuity property:
Lemma 2.5.
Let ( x, q ) ∈ J and suppose q is a non-integer. Then (i) for each positive integer m there exists a neighborhood W ⊂ R of ( x, q ) such that (2.2) b ( y, r ) . . . b m ( y, r ) ≤ b ( x, q ) . . . b m ( x, q ) for all ( y, r ) ∈ W ∩ J ;(ii) if ( y n , r n ) converges to ( x, q ) in J from above, then ( b i ( y n , r n )) convergesto ( b i ( x, q )) .Proof. (i) By the definition of greedy expansions we have n X i =1 b i ( x, q ) q i > x − q n whenever b n ( x, q ) < α ( q ) . If ( y, r ) ∈ J is sufficiently close to ( x, q ), then A q = A r , α ( r ) = α ( q ), and n X i =1 b i ( x, q ) r i > y − r n whenever n ≤ m and b n ( x, q ) < α ( r ) . These inequalities imply (2.2).(ii) If y n ≥ x and r n ≥ q , we deduce from the definition of the greedy expansionthat ( b i ( x, q )) ≤ ( b i ( y n , r n ))for every n . Equivalently, we have b ( x, q ) . . . b m ( x, q ) ≤ b ( y n , r n ) . . . b m ( y n , r n )for all positive integers m and n . It remains to notice that by the previous part theconverse inequality also holds for each fixed m if n is large enough. (cid:3) From Lemmas 2.3 and 2.5 we deduce the following result:
Proposition 2.6.
Consider ( x, q ) ∈ J with a non-integer base q and assume thatthe greedy expansion ( b i ( x, q )) is infinite. If ( y n , r n ) converges to ( x, q ) in J , thenboth ( a i ( y n , r n )) and ( b i ( y n , r n )) converge to ( b i ( x, q )) = ( a i ( x, q )) .Proof. For each positive integer m there exists a neighborhood W ⊂ R of ( x, q )such that for all ( y, r ) ∈ W ∩ J , a ( x, q ) . . . a m ( x, q ) ≤ a ( y, r ) . . . a m ( y, r ) ≤ b ( y, r ) . . . b m ( y, r ) ≤ b ( x, q ) . . . b m ( x, q ) . The result follows from our assumption that ( a i ( x, q )) = ( b i ( x, q )). (cid:3) Theorem 2.7.
Let q > be a real number. Then (i) for each r ∈ (1 , q ) , the Hausdorff dimension of the set G r,q := ( ∞ X i =1 b i ( x, r ) q i : x ∈ J r ) equals log r/ log q ; TWO-DIMENSIONAL UNIVOQUE SET 5 (ii) the set G q := [ { G r,q : r ∈ (1 , q ) } is of first category, has Lebesgue measure zero and Hausdorff dimensionone.Proof. (i) It is well known ([15], [16]) and easy to prove that the set of numbers r > β i ( r )) is finite is dense in [1 , ∞ ). Moreover, if ( β i ( r )) is finiteand β n ( r ) is its last nonzero element, then ( α i ( r )) = ( β ( r ) . . . β n − ( r ) β − n ( r )) ∞ ( β − n ( r ) := β n ( r ) − G s,q ⊂ G t,q whenever 1 < s < t < q . Hence it is enough to prove that dim H G r,q = log r/ log q for those values r ∈ (1 , q ) for which ( α i ( r )) is periodic.Fix r ∈ (1 , q ) such that ( α i ) := ( α i ( r )) is periodic and let n ∈ N be such that( α i ) = ( α . . . α n ) ∞ . Let us denote by W r the set consisting of the finite words w ij := α . . . α j − i, ≤ i < α j , ≤ j ≤ n and w α n n := α . . . α n − α n . Let F ′ r be the set of sequences ( c i ) = c c such that for each k ≥ c k +1 . . . c k + n ≤ α . . . α n holds. Note that the set F ′ r consists of those sequences( c i ) such that each tail of ( c i ) (including ( c i ) itself) starts with a word belonging to W r . It follows from Propositions 2.1 and 2.4 that a sequence ( b i ) is greedy in base r if and only if b m ∈ A r for all m ≥ b m + k +1 b m + k +2 . . . < α α . . . for all k ≥
0, whenever b m < α . Therefore, any greedy expansion ( b i ) = α ∞ in base r can be written as α ℓ c c . . . for some ℓ ≥ α denotes the empty word) and some sequence ( c i ) belonging to F ′ r . Conversely, if no tail of a sequence belonging to F ′ r equals ( α i ), then it is thegreedy expansion in base r of some x ∈ J r . Hence if we set F r,q := ( ∞ X i =1 c i q i : ( c i ) ∈ F ′ r ) , then F r,q \ G r,q is countable and G r,q can be covered by countably many sets similarto F r,q . Since the union of countably many sets of Hausdorff dimension s is still ofHausdorff dimension s , we have dim H G r,q = dim H F r,q .We associate with each word w ij ∈ W r a similarity S ij : J q → J q defined by theformula S ij ( x ) := α q + · · · + α j − q j − + iq j + xq j , x ∈ J q . It follows from Proposition 2.1 and the definition of F r,q that(2.3) F r,q = [ S ij ( F r,q )where the union runs over all i and j for which w ij ∈ W r . Applying Proposition 2.1again, it follows that r is the largest element of the set of numbers t > α i ( t ) = α i , 1 ≤ i ≤ n . Hence α . . . α n < α ( q ) . . . α n ( q ) and therefore eachsequence in F ′ r is the greedy expansion in base q of some x ∈ F r,q . It follows thatthe sets S ij ( F r,q ) on the right side of (2.3) are disjoint. Moreover, the function x ( b i ( x, q )) that maps F r,q onto F ′ r is increasing. Using the definition of F ′ r itis easily seen that the limit of each monotonic sequence of elements in F r,q belongsto F r,q . We conclude that the closed set F r,q is the (nonempty compact) invariantset of this system of similarities. An application of Propositions 9.6 and 9.7 in [9]yields that dim H F r,q = dim H G r,q = s MARTIJN DE VRIES AND VILMOS KOMORNIK where s is the real solution of the equation α q s + · · · + α n − q ( n − s + α n + 1 q ns = 1 . Since α r + · · · + α n − r n − + α n + 1 r n = 1we have s = log r/ log q .(ii) It follows at once from Theorem 2.7(i) that dim H G q = 1. Let r ∈ (1 , q ) besuch that ( α i ( r )) is periodic. The proof of Theorem 2.7(i) shows that G r,q ⊂ ∞ [ n =1 ( a n + b n F r,q )for some constants a n , b n ∈ R ( n ∈ N ). Since F r,q is a closed set of Hausdorffdimension less than one, it follows in particular that the sets a n + b n F r,q are nowheredense null sets. Since G s,q ⊂ G t,q whenever 1 < s < t < q , the set G q is a null setof first category. (cid:3) Theorem 2.8.
Let q > be a real number. (i) Let v := b ℓ +1 ( y, q ) . . . b ℓ + m ( y, q ) for some y ∈ [0 , and some integers ℓ ≥ and m ≥ . The set Y v of numbers x ∈ J q for which the word v does notoccur in the greedy expansion of x in base q has Hausdorff dimension lessthan one. (ii) The set Y of numbers x ∈ J q for which at least one word of the form b ℓ +1 ( y, q ) . . . b ℓ + m ( y, q ) ( ℓ ≥ , m ≥ , y ∈ [0 , does not occur in thegreedy expansion of x in base q is of first category, has Lebesgue measurezero and Hausdorff dimension one.Proof. (i) Using the inequality ( b i ( y, q )) < ( α i ( q )), it follows from Proposition 2.4that for some k ∈ N , there exist positive integers m , . . . , m k and nonnegativeintegers ℓ , . . . , ℓ k satisfying α m j ( q ) > ℓ j < α m j ( q ) for each 1 ≤ j ≤ k , suchthat v is a subword of w := α ( q ) . . . α m − ( q ) ℓ . . . α ( q ) . . . α m k − ( q ) ℓ k . Let W q and F ′ q be the same as the sets W r and F ′ r defined in the proof of theprevious theorem, but now with ( α i ) := ( α i ( q )) and n ≥ max { m , . . . , m k } largeenough such that the inequality(2.4) (cid:18) q n (cid:19) k < q m + ··· + m k holds. If w i j , . . . , w i k j k are k words belonging to W q such that i j . . . i k j k = ℓ m . . . ℓ k m k , we associate with them a similarity S i j ...i k j k : J q → J q defined by the formula S i j ...i k j k ( x ) = α q + · · · + α j − q j − + i q j + α q j +1 + · · · + α j − q j + j − + i q j + j ...+ α q j + ··· + j k − +1 + · · · + α j k − q j + ··· + j k − + i k q j + ··· + j k + xq j + ··· + j k , x ∈ J q . TWO-DIMENSIONAL UNIVOQUE SET 7
Let G ′ q denote the set of those sequences belonging to F ′ q which do not contain theword w , and let G q := ( ∞ X i =1 c i q i : ( c i ) ∈ G ′ q ) . Since ( α i ) = ( α i ( q )), a sequence belonging to F ′ q is not necessarily the greedyexpansion in base q of a number x ∈ J q , but this does not affect our proof. It isimportant, however, that any greedy expansion ( b i ) = α ∞ in base q can be writtenas α ℓ c c . . . for some ℓ ≥ c i ) belonging to F ′ q . If Y w denotesthe set of numbers x ∈ J q for which the word w does not occur in ( b i ( x, q )) thenthe latter fact implies that the set Y w \ { α / ( q − } can be covered by countablymany sets similar to G q .It follows from the definition of G q that G q ⊂ [ S i j ...i k j k ( G q )where the union runs over all i j . . . i k j k for which the similarity S i j ...i k j k isdefined above. Hence G q ⊂ [ S i j ...i k j k ( G q )and thus G q ⊂ H q where H q is the (nonempty compact) invariant set of this systemof similarities. Let ˜ α i := α i for 1 ≤ i < n and ˜ α n := α n + 1. From Proposition 9.6in [9] we know that dim H H q ≤ s where s is the real solution of the equation(2.5) n X j =1 n X j =1 · · · n X j k =1 (cid:18) Π ki =1 ˜ α j i q ( j + ··· + j k ) s (cid:19) − q ( m + ··· + m k ) s = 1 . Denoting the left side of (2.5) by C ( s ), we have C (1) + 1 q m + ··· + m k = n X i =1 ˜ α i q i ! k < (cid:18) q n (cid:19) k . By (2.4) we have C (1) <
1, and thus dim H Y v ≤ dim H Y w ≤ dim H H q < Y w (and thus Y v ) is of first category. Since Y is a countable union ofsets of the form Y v it follows that Y is a null set of first category. Let r ∈ (1 , q )and let G r,q be the set defined in Theorem 2.7. Due to Theorem 2.7(i) it is nowsufficient to show that G r,q ⊂ Y . To this end, choose a number p ∈ ( r, q ), and let n ∈ N be large enough such that the inequalities α ( r ) . . . α n ( r ) < α ( p ) . . . α n ( p ) < α ( q ) . . . α n ( q )hold. Note that such an integer n exists by Proposition 2.1. From Propositions 2.1and 2.4 we conclude that the sequence 0 α ( p ) . . . α n ( p )0 ∞ equals ( b i ( y, q )) for some y ∈ [0 ,
1) while the word 0 α ( p ) . . . α n ( p ) does not occur in the greedy expansionin base r of any number x ∈ J r . (cid:3) Proof of Theorem 1.2
The following characterization of unique expansions readily follows from Propo-sition 2.4.
Proposition 3.1.
Fix q > . A sequence ( c i ) of integers c i ∈ A q is the uniqueexpansion of some x ∈ J q if and only if c n +1 c n +2 . . . < α ( q ) α ( q ) . . . whenever c n < α ( q ) and c n +1 c n +2 . . . < α ( q ) α ( q ) . . . whenever c n > . MARTIJN DE VRIES AND VILMOS KOMORNIK
In what follows we use the notations ( a i ( x, q )), ( b i ( x, q )), ( α i ( q )) and ( β i ( q ))as introduced in Section 1. If x and q are clear from the context, then we omitthese arguments and we simply write a i , b i , α i and β i . If two couples ( x, q ) and( x ′ , q ′ ) are considered simultaneously, then we also write a ′ i , b ′ i , α ′ i and β ′ i insteadof a i ( x ′ , q ′ ), b i ( x ′ , q ′ ), α i ( q ′ ) and β i ( q ′ ). Lemma 3.2.
Given ( x, q ) ∈ J , the following two conditions are equivalent: a n +1 a n +2 . . . ≤ α α . . . whenever a n > a n +1 a n +2 . . . ≤ β β . . . whenever a n > . Proof.
Since ( α i ) ≤ ( β i ), it suffices to show that if there exists a positive integer n such that a n > a n +1 a n +2 . . . > α α . . . , then there exists also a positive integer n ′ such that a n ′ > a n ′ +1 a n ′ +2 . . . > β β . . . . If the greedy expansion ( β i ) is infinite, then ( β i ) = ( α i ) and we may choose n ′ = n .If ( β i ) has a last nonzero digit β ℓ , then ( α i ) = ( α . . . α ℓ ) ∞ with α . . . α ℓ − α ℓ = β . . . β ℓ − β − ℓ ( β − ℓ := β ℓ − α ℓ < α . Since we have a n +1 a n +2 . . . > ( α . . . α ℓ ) ∞ by assumption, there exists a nonnegative integer j satisfying a n +1 . . . a n + jℓ = ( α . . . α ℓ ) j and a n + jℓ +1 . . . a n +( j +1) ℓ > α . . . α ℓ . Putting n ′ := n + jℓ it follows that a n ′ > a n ′ +1 . . . a n ′ + ℓ ≥ β . . . β ℓ . It follows from our assumption a n +1 a n +2 . . . > α α . . . that ( α i ) < α ∞ and ( a i ) = α ∞ . It follows from Proposition 2.2 that ( a i ) has no tail equal to α ∞ , so that a n ′ + ℓ +1 a n ′ + ℓ +2 . . . > ∞ . We conclude that a n ′ +1 a n ′ +2 . . . > β β . . . . (cid:3) Definition.
We say that ( x, q ) ∈ J belongs to the set V if one of the two equivalentconditions of the preceding lemma is satisfied. Moreover, we define V q := { x ∈ J q : ( x, q ) ∈ V } , q > . It follows from Proposition 3.1 that U ⊂ V ⊂ J . Proof of Theorem 1.2.
We need to prove that U ∩ J = V .First we show that V ⊂ U . In order to do so, we introduce for each fixed q > U ′ q and V ′ q , defined by U ′ q := { ( a i ( x, q )) : x ∈ U q } and V ′ q := { ( a i ( x, q )) : x ∈ V q } . Observe that U ′ q is simply the set of unique expansions in base q . It follows easilyfrom Propositions 2.1, 2.2 and 3.1 that U ′ q ⊂ V ′ q for each q >
1, and that V ′ q ⊂ U ′ r for each r > q such that ⌈ q ⌉ = ⌈ r ⌉ . Since we also have U q = V q = [0 ,
1] if q > U ∩ J ⊂ V . Since U ⊂ V it is sufficient to prove that if( x, q ) ∈ J \ V , then ( x ′ , q ′ ) / ∈ V for all ( x ′ , q ′ ) ∈ J close enough to ( x, q ). ApplyingLemma 3.2 there exist two positive integers n and m such that(3.1) a n > a n +1 . . . a n + m > β . . . β m . TWO-DIMENSIONAL UNIVOQUE SET 9
This implies in particular that q is not an integer, because otherwise ( α i ) = ( β i ) = β ∞ . Hence, if q ′ is sufficiently close to q , then(3.2) β ′ . . . β ′ m ≤ β . . . β m by Lemma 2.5. It follows from the definition of quasi-greedy expansions that a q + · · · + a j − q j − + a + j q j + 1 q j + m > x whenever a j < α ,where a + j := a j + 1. If ( x ′ , q ′ ) ∈ J is sufficiently close to ( x, q ), then α = α ′ , theinequality (3.2) is satisfied, a ′ . . . a ′ n + m ≥ a . . . a n + m by Lemma 2.3(i), and(3.3) a q ′ + · · · + a j − ( q ′ ) j − + a + j ( q ′ ) j + 1( q ′ ) j + m > x ′ whenever j ≤ n + m and a j < α .Now we distinguish between two cases.If a ′ . . . a ′ n + m = a . . . a n + m , then we have a ′ n > a ′ n +1 . . . a ′ n + m > β . . . β m ≥ β ′ . . . β ′ m by (3.1) and (3.2). This proves that ( x ′ , q ′ ) / ∈ V .If a ′ . . . a ′ n + m > a . . . a n + m , then let us consider the smallest j for which a ′ j > a j .It follows from (3.2) and (3.3) that a ′ j = a + j > a ′ j +1 . . . a ′ j + m = β m > β . . . β m ≥ β ′ . . . β ′ m . Hence ( x ′ , q ′ ) / ∈ V again. (cid:3) Remark.
It is the purpose of this remark to describe the set U \ J . For each m ∈ N ,we define the number q m ∈ ( m, m + 1) by the equation1 = mq m + 1 q m . Fix q ∈ ( m, q m ]. Since α ( q ) = m and α ( q ) = 0, Proposition 3.1 implies that asequence ( c i ) ∈ { , . . . , m } N belongs to U ′ q if and only if for each n ∈ N , we have c n < m = ⇒ c n +1 < m and c n > ⇒ c n +1 > . Denoting the set of all such sequences by D ′ m and putting for m > D ′ = { ∞ , ∞ } ), D m := ( ∞ X i =1 c i m i : ( c i ) ∈ D ′ m ) , one may verify that U \ J = { (0 , } ∪ ∞ [ m =2 ( D m \ [0 , × { m } . Proof of Theorem 1.1
We need some results on the Hausdorff dimension of the sets U q and V q for q >
1. It follows from Theorem 1.2 that U q ⊂ U q ⊂ V q . Moreover, if an element x ∈ V q \ U q has an infinite greedy expansion in base q , then ( b i ( x, q )) must end with α ( q ) α ( q ) . . . as follows from Propositions 2.4 and 3.1; hence V q \ U q is countableand the sets U q , U q and V q have the same Hausdorff dimension for each q > Proposition 4.1.
We have (i) lim q ↑ dim H U q = 1 ; (ii) dim H U q < for all non-integer q > .Proof. (i) Assume that q ∈ (1 ,
2) is larger than the tribonacci number, i.e.,1 q + 1 q + 1 q < , and let N = N ( q ) ≥ q + · · · + 1 q N − < . Hence α ( q ) = · · · = α N − ( q ) = 1. Let us denote by I q the set of numbers x ∈ J q which have an expansion ( c i ) satisfying 0 < c kN +1 + · · · + c kN + N < N for everynonnegative integer k . Since in such expansions ( c i ), a zero (one) is followed byat most 2 N − I q ⊂ U q . Moreover, the set I q is closed and thus compact. In order to prove this,observe first that the set I q is closed from above by virtue of Lemma 2.5(ii).Hence I q is closed because I q is symmetric relative to J q .It suffices now to prove that(4.1) dim H I q = log(2 N − N log q ;indeed, q ↑ N → ∞ , hence dim H I q → H U q → I q = [ S c ...c N ( I q )where the union runs over the words c . . . c N of length N consisting of zeros andones satisfying 0 < c + · · · + c N < N , and S c ...c N : J q → J q is given by S c ...c N ( x ) := (cid:16) c q + · · · + c N q N (cid:17) + xq N , x ∈ J q . In other words, I q is the (nonempty compact) invariant set of the iterated functionsystem formed by these 2 N − S c ...c N ( I q ) on the right sideof (4.2) are disjoint because S c ...c N ( I q ) ⊂ I q ⊂ U q , and since all similarity ratiosare equal to q − N , it follows from Propositions 9.6 and 9.7 in [9] that the Hausdorffdimension s of I q is the real solution of the equation(2 N − q − Ns = 1 , which is equivalent to (4.1).(ii) Let q > n ∈ N be such that α n ( q ) < α ( q ). Itfollows from Proposition 3.1 that the word 1(0) n does not occur in ( b i ( x, q )) if x belongs to U q . Applying Theorem 2.8(i) with y = q − , ℓ = 0 and m = n + 1, weconclude that dim H U q < (cid:3) Proof of Theorem 1.1. (ii) Let q > V q \ U q is countable,Proposition 4.1(ii) yields that dim H V q <
1. This implies in particular that the set V q is a one-dimensional null set. Applying Theorem 1.2 (and the remark followingits proof) and Fubini’s theorem we conclude that U is a two-dimensional null set.(i) Since U q is not closed for all q > U cannot be closed. Since U is a two-dimensional null set, it has no interior points. It remains to show that U (and thus U ) has no isolated points. If q > U q is dense We call a set X ⊂ R closed from above if x belongs to X whenever there exists a sequence( x n ) of elements of X that converges to x from above. TWO-DIMENSIONAL UNIVOQUE SET 11 in J q = [0 , q > x, q ) ∈ U is not isolated because U ′ q ⊂ U ′ r whenever q < r and ⌈ q ⌉ = ⌈ r ⌉ .(iii) From Corollary 7.10 in [9] we may conclude that for almost all q > H U q ≤ max { , dim H U − } which would contradict Proposition 4.1(i) if we had dim H U < (cid:3) Acknowledgements.
The first author has been supported by NWO, Project nr.ISK04G. Part of this work was done during a visit of the second author at theDepartment of Mathematics of the Delft University of Technology. He is gratefulfor this invitation and for the excellent working conditions.
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On the uniqueness of the expansions P q − n i , ActaMath. Hungar. (1991), no. 3–4, 333–342.[7] P. Erd˝os, I. Jo´o, V. Komornik, Characterization of the unique expansions P ∞ i =1 q − n i and related problems , Bull. Soc. Math. France (1990), no. 3, 377–390.[8] P. Erd˝os, I. Jo´o, V. Komornik, On the number of q -expansions , Ann. Univ. Sci. Budapest.E¨otv¨os Sect. Math. (1994), 109–118.[9] K. Falconer, Fractal Geometry. Mathematical Foundations and Applications , John Wiley& Sons, Chicester, second edition, 2003.[10] P. Glendinning, N. Sidorov,
Unique representations of real numbers in non-integer bases ,Math. Res. Lett. (2001), no. 4, 535–543.[11] G. Kall´os, The structure of the univoque set in the small case , Publ. Math. Debrecen (1999), no. 1–2, 153–164.[12] G. Kall´os, The structure of the univoque set in the big case , Publ. Math. Debrecen (2001), no. 3–4, 471–489.[13] I. K´atai, G. Kall´os, On the set for which is univoque , Publ. Math. Debrecen (2001),no. 4, 743–750.[14] V. Komornik, P. Loreti, Unique developments in non-integer bases , Amer. Math. Monthly (1998), no. 7, 636–639.[15] V. Komornik, P. Loreti,
On the topological structure of univoque sets , J. Number Theory, (2007), no. 1, 157–183.[16] W. Parry,
On the β -expansions of real numbers , Acta Math. Acad. Sci. Hungar. (1960),401–416.[17] A. R´enyi, Representations for real numbers and their ergodic properties , Acta Math. Acad.Sci. Hungar. (1957), 477-493.[18] N. Sidorov, Almost every number has a continuum of β -expansions , Amer. Math. Monthly (2003), no. 9, 838–842.[19] N. Sidorov, Combinatorics of linear iterated function systems with overlaps , Nonlinearity (2007), 1299–1312. Delft University of Technology, Mekelweg 4, 2628 CD Delft, the Netherlands
E-mail address : [email protected] D´epartement de Math´ematique, Universit´e de Strasbourg, 7 rue Ren´e Descartes,67084 Strasbourg Cedex, France
E-mail address : [email protected] r X i v : . [ m a t h . N T ] D ec A TWO-DIMENSIONAL UNIVOQUE SET
MARTIJN DE VRIES AND VILMOS KOMORNIK
Abstract.
Let J ⊂ R be the set of couples ( x, q ) with q > x hasat least one representation of the form x = P ∞ i =1 c i q − i with integer coefficients c i satisfying 0 ≤ c i < q , i ≥
1. In this case we say that ( c i ) = c c . . . is anexpansion of x in base q . Let U be the set of couples ( x, q ) ∈ J such that x has exactly one expansion in base q .In this paper we deduce some topological and combinatorial properties ofthe set U . We characterize the closure of U , and we determine its Hausdorffdimension. For ( x, q ) ∈ J , we also prove new properties of the lexicographicallylargest expansion of x in base q . Introduction
Let J be the set consisting of all elements ( x, q ) ∈ R × (1 , ∞ ) such that thereexists at least one sequence ( c i ) = c c . . . of integers satisfying 0 ≤ c i < q for all i ,and(1.1) x = c q + c q + · · · . If (1.1) holds, we say that ( c i ) is an expansion of x in base q , and if the base q isunderstood from the context, we sometimes simply say that ( c i ) is an expansion of x . The numbers c i of an expansion ( c i ) are usually referred to as digits . We denoteby ⌈ q ⌉ the smallest integer larger than or equal to q . The alphabet A q is the set of“admissible” digits in base q , i.e., A q = { , . . . , ⌈ q ⌉ − } .If q > ≤ x ≤ ( ⌈ q ⌉ − / ( q − x inbase q , the so-called quasi-greedy expansion ( a i ( x, q )), may be defined recursivelyas follows. For x = 0 we set ( a i ( x, q )) := 0 ∞ . If x > a i ( x, q ) has already beendefined for 1 ≤ i < n (no condition if n = 1), then a n ( x, q ) is the largest elementof A q satisfying a ( x, q ) q + · · · + a n ( x, q ) q n < x. One easily verifies that ( a i ( x, q )) is indeed an expansion of x in base q . Therefore( x, q ) ∈ J ⇐⇒ q > x ∈ J q := (cid:20) , ⌈ q ⌉ − q − (cid:21) . Let us denote by U the set of couples ( x, q ) ∈ J such that x has exactly one expansion in base q . For example, (0 , q ) ∈ U for every q >
1, but U has manymore elements. The main purpose of this paper is to describe the topological andcombinatorial nature of U . We will prove the following theorem: Theorem 1.1. (i)
The set U is not closed. Its closure U is a Cantor set . Date : October 13, 2018.2000
Mathematics Subject Classification.
Primary:11A63, Secondary:11B83.
Key words and phrases.
Greedy expansion, beta-expansion, univoque sequence, univoque set,Cantor set, Hausdorff dimension. We recall that a Cantor set is a nonempty closed set having neither interior nor isolatedpoints. (ii)
Both U and U are two-dimensional Lebesgue null sets. (iii) Both U and U have Hausdorff dimension two. As far as we know this two-dimensional univoque set has not yet been inves-tigated. There exists, however, a number of papers devoted to the study of itsone-dimensional sections U := { q > , q ) ∈ U } and U q := { x ∈ J q : ( x, q ) ∈ U } , q > . The study of U started with the paper of Erd˝os, Horv´ath and Jo´o [6] and wasstudied subsequently in [4], [5], [7], [8], [15], [16], [17]. We recall in particular that U and its closure U have Lebesgue measure zero and Hausdorff dimension one.The sets U q have been investigated in [3], [4], [5], [11], [13], [14]. It is known (see[5]) that U q is closed if and only if q does not belong to the null set U , and that itsclosure U q has Lebesgue measure zero for all non-integer bases q >
1. Moreover,the set of numbers x ∈ J q having a continuum of expansions in base q has fullLebesgue measure for each non-integer q > U byusing the quasi-greedy expansions ( a i ( x, q )). We write for brevity α i ( q ) := a i (1 , q ), i ∈ N := { , , . . . } , q >
1. Note that α ( q ) = ⌈ q ⌉ −
1, the largest admissible digitin base q . In the statement of the following theorem we use the lexicographic orderbetween sequences and we define the conjugate (in base q ) of a digit c ∈ A q by c := α ( q ) − c . If c i ∈ A q , i ≥
1, we shall also write c . . . c n instead of c . . . c n and c c . . . instead of c c . . . . Theorem 1.2.
A point ( x, q ) ∈ J belongs to U if and only if a n +1 ( x, q ) a n +2 ( x, q ) . . . ≤ α ( q ) α ( q ) . . . whenever a n ( x, q ) > . Along with the quasi-greedy expansion, we also need the notion of the greedyexpansion ( b i ( x, q )) for x ∈ J q , introduced by R´enyi [19]. It can be defined by aslight modification of the above recursion: if b i ( x, q ) has already been defined forall 1 ≤ i < n (no condition if n = 1), then b n ( x, q ) is the largest element of A q satisfying b ( x, q ) q + · · · + b n ( x, q ) q n ≤ x. Note that the greedy expansion ( b i ( x, q )) of a number x ∈ J q is the lexicographicallylargest expansion of x in base q . We denote the greedy expansion of 1 in base q by( β i ( q )) := ( b i (1 , q )).The rest of this paper is organized as follows. In the next section we give a shortoverview of some basic results on greedy and quasi-greedy expansions, and we provesome new results concerning the coordinate-wise convergence of sequences of theseexpansions. We shall prove (see Theorem 2.7) that the set of numbers x ∈ J q forwhich the greedy expansion of x in base q is not the greedy expansion of a numberbelonging to J p in any smaller base p ∈ (1 , q ) is of full Lebesgue measure andits complement in J q is a set of first category and Hausdorff dimension one. Weshall also prove (see Theorem 2.8) that for each word v := b ℓ +1 ( x, q ) . . . b ℓ + m ( x, q )( ℓ ≥ , m ≥ , x ∈ [0 , Y v ⊂ J q of first category and Hausdorffdimension less than one, such that the word v occurs in the greedy expansion inbase q of every number belonging to J q \ Y v . Using (some of) the results of Section 2we prove Theorem 1.2 in Section 3 and Theorem 1.1 in Section 4. TWO-DIMENSIONAL UNIVOQUE SET 3 Greedy and quasi-greedy expansions
In this paper we consider only one-sided sequences of nonnegative integers. Weequip this set of sequences { , , . . . } N with the topology of coordinate-wise con-vergence. We say that an expansion is infinite if it has infinitely many nonzeroelements; otherwise it is called finite . Using this terminology, the quasi-greedy ex-pansion ( a i ( x, q )) of a number x ∈ J q \ { } is the lexicographically largest infinite expansion of x in base q . Moreover, if the greedy expansion of x ∈ J q is infinite,then ( a i ( x, q )) = ( b i ( x, q )).The family of all quasi-greedy expansions is characterized by the following propo-sitions (see [1] or [5] for a proof): Proposition 2.1.
The map q ( α i ( q )) is an increasing bijection from the openinterval (1 , ∞ ) onto the set of all infinite sequences ( α i ) satisfying α k +1 α k +2 . . . ≤ α α . . . for all k ≥ . Proposition 2.2.
For each q > , the map x ( a i ( x, q )) is an increasing bijectionfrom J q \ { } onto the set of all infinite sequences ( a i ) satisfying a n ∈ A q for all n ≥ and a n +1 a n +2 . . . ≤ α ( q ) α ( q ) . . . whenever a n < α ( q ) . The quasi-greedy expansions have a lower semicontinuity property for the ordertopology induced by the lexicographic order. More precisely, we have the followingresult.
Lemma 2.3.
Let ( x, q ) ∈ J and ( y n , r n ) ∈ J , n ∈ N . Then (i) for each positive integer m there exists a neighborhood W ⊂ R of ( x, q ) such that (2.1) a ( y, r ) . . . a m ( y, r ) ≥ a ( x, q ) . . . a m ( x, q ) for all ( y, r ) ∈ W ∩ J ;(ii) if y n ↑ x and r n ↑ q , then ( a i ( y n , r n )) converges to ( a i ( x, q )) .Proof. (i) We may assume that x = 0. By definition of the quasi-greedy expansionwe have n X i =1 a i ( x, q ) q i < x for all n = 1 , , . . . . For any fixed positive integer m , if ( y, r ) ∈ J is sufficiently close to ( x, q ), then r > ⌈ q ⌉ −
1, i.e., A q ⊂ A r , and n X i =1 a i ( x, q ) r i < y, n = 1 , . . . , m. These inequalities imply (2.1).(ii) If y n ≤ x and r n ≤ q , we deduce from the definition of the quasi-greedyexpansion that ( a i ( x, q )) ≥ ( a i ( y n , r n ))for every n . Equivalently, we have a ( x, q ) . . . a m ( x, q ) ≥ a ( y n , r n ) . . . a m ( y n , r n )for all positive integers m and n . It remains to notice that by the previous part theconverse inequality also holds for each fixed m if n is large enough. (cid:3) The family of greedy expansions has already been characterized by Parry [18]:
MARTIJN DE VRIES AND VILMOS KOMORNIK
Proposition 2.4.
For a given base q > , the map x ( b i ( x, q )) is an increasingbijection from J q onto the set of all sequences ( b i ) satisfying b n ∈ A q for all n ≥ and b n +1 b n +2 . . . < α ( q ) α ( q ) . . . whenever b n < α ( q ) . The greedy expansions have the following upper semicontinuity property:
Lemma 2.5.
Let ( x, q ) ∈ J , ( y n , r n ) ∈ J , n ∈ N and suppose q / ∈ N . Then (i) for each positive integer m there exists a neighborhood W ⊂ R of ( x, q ) such that (2.2) b ( y, r ) . . . b m ( y, r ) ≤ b ( x, q ) . . . b m ( x, q ) for all ( y, r ) ∈ W ∩ J ;(ii) if y n ↓ x and r n ↓ q , then ( b i ( y n , r n )) converges to ( b i ( x, q )) .Proof. (i) By the definition of greedy expansions we have n X i =1 b i ( x, q ) q i > x − q n whenever b n ( x, q ) < α ( q ) . If ( y, r ) ∈ J is sufficiently close to ( x, q ), then A r = A q , α ( r ) = α ( q ), and n X i =1 b i ( x, q ) r i > y − r n whenever n ≤ m and b n ( x, q ) < α ( r ) . These inequalities imply (2.2).(ii) If y n ≥ x and r n ≥ q , we deduce from the definition of the greedy expansionthat ( b i ( x, q )) ≤ ( b i ( y n , r n ))for every n . Equivalently, we have b ( x, q ) . . . b m ( x, q ) ≤ b ( y n , r n ) . . . b m ( y n , r n )for all positive integers m and n . It remains to notice that by the previous part theconverse inequality also holds for each fixed m if n is large enough. (cid:3) From Lemmas 2.3 and 2.5 we deduce the following result:
Proposition 2.6.
Consider ( x, q ) ∈ J with a non-integer base q and assume thatthe greedy expansion ( b i ( x, q )) is infinite. If ( y n , r n ) converges to ( x, q ) in J , thenboth ( a i ( y n , r n )) and ( b i ( y n , r n )) converge to ( b i ( x, q )) = ( a i ( x, q )) .Proof. For each positive integer m there exists a neighborhood W ⊂ R of ( x, q )such that for all ( y, r ) ∈ W ∩ J , a ( x, q ) . . . a m ( x, q ) ≤ a ( y, r ) . . . a m ( y, r ) ≤ b ( y, r ) . . . b m ( y, r ) ≤ b ( x, q ) . . . b m ( x, q ) . The result follows from our assumption that ( a i ( x, q )) = ( b i ( x, q )). (cid:3) Theorem 2.7.
Let q > be a real number. Then (i) for each r ∈ (1 , q ) , the Hausdorff dimension of the set G r,q := ( ∞ X i =1 b i ( x, r ) q i : x ∈ J r ) equals log r/ log q ; TWO-DIMENSIONAL UNIVOQUE SET 5 (ii) the set G q := [ { G r,q : r ∈ (1 , q ) } is of first category, has Lebesgue measure zero and Hausdorff dimensionone.Proof. (i) It is well known (see, e.g., [17], [18]) and easy to prove that the set ofnumbers r > β i ( r )) is finite is dense in [1 , ∞ ). Moreover, if ( β i ( r )) is fi-nite and β n ( r ) is its last nonzero element, then ( α i ( r )) = ( β ( r ) . . . β n − ( r ) β − n ( r )) ∞ ( β − n ( r ) := β n ( r ) − G s,q ⊂ G t,q whenever 1 < s < t < q . Hence it is enough to prove that dim H G r,q = log r/ log q for those values r ∈ (1 , q ) for which ( α i ( r )) is periodic.Fix r ∈ (1 , q ) such that ( α i ) := ( α i ( r )) is periodic and let n ∈ N be such that( α i ) = ( α . . . α n ) ∞ . Let us denote by W r the set consisting of the finite words w ij := α . . . α j − i, ≤ i < α j , ≤ j ≤ n and w α n n := α . . . α n − α n . Let F ′ r be the set of sequences ( c i ) = c c such that for each k ≥ c k +1 . . . c k + n ≤ α . . . α n holds. Note that the set F ′ r consists of those sequences( c i ) such that each tail of ( c i ) (including ( c i ) itself) starts with a word belonging to W r . It follows from Propositions 2.1 and 2.4 that a sequence ( b i ) is greedy in base r if and only if b m ∈ A r for all m ≥ b m + k +1 b m + k +2 . . . < α α . . . for all k ≥
0, whenever b m < α . Therefore, any greedy expansion ( b i ) = α ∞ in base r can be written as α ℓ c c . . . for some ℓ ≥ α denotes the empty word) and some sequence ( c i ) belonging to F ′ r . Conversely, if no tail of a sequence belonging to F ′ r equals ( α i ), then it is thegreedy expansion in base r of some x ∈ J r . Hence if we set F r,q := ( ∞ X i =1 c i q i : ( c i ) ∈ F ′ r ) , then F r,q \ G r,q is countable and G r,q can be covered by countably many sets similarto F r,q . Since the union of countably many sets of Hausdorff dimension s is still ofHausdorff dimension s , we have dim H G r,q = dim H F r,q .We associate with each word w ij ∈ W r a similarity S ij : J q → J q defined by theformula S ij ( x ) := α q + · · · + α j − q j − + iq j + xq j , x ∈ J q . It follows from Proposition 2.1 and the definition of F r,q that(2.3) F r,q = [ S ij ( F r,q )where the union runs over all i and j for which w ij ∈ W r . Applying Proposition 2.1again, it follows that r is the largest element of the set of numbers t > α i ( t ) = α i , 1 ≤ i ≤ n . Hence α . . . α n < α ( q ) . . . α n ( q ) and therefore eachsequence in F ′ r is the greedy expansion in base q of some x ∈ F r,q . It follows thatthe sets S ij ( F r,q ) on the right side of (2.3) are disjoint. Moreover, the function x ( b i ( x, q )) that maps F r,q onto F ′ r is increasing. Using the definition of F ′ r itis easily seen that the limit of each monotonic sequence of elements in F r,q belongsto F r,q . We conclude that the closed set F r,q is the (nonempty compact) invariantset of this system of similarities. An application of Propositions 9.6 and 9.7 in [9]yields that dim H F r,q = dim H G r,q = s MARTIJN DE VRIES AND VILMOS KOMORNIK where s is the real solution of the equation α q s + · · · + α n − q ( n − s + α n + 1 q ns = 1 . Since α r + · · · + α n − r n − + α n + 1 r n = 1we have s = log r/ log q .(ii) It follows at once from Theorem 2.7(i) that dim H G q = 1. Let r ∈ (1 , q ) besuch that ( α i ( r )) is periodic. The proof of Theorem 2.7(i) shows that G r,q ⊂ ∞ [ n =1 ( a n + b n F r,q )for some constants a n , b n ∈ R ( n ∈ N ). Since F r,q is a closed set of Hausdorffdimension less than one, it follows in particular that the sets a n + b n F r,q are nowheredense null sets. Since G s,q ⊂ G t,q whenever 1 < s < t < q , the set G q is a null setof first category. (cid:3) Theorem 2.8.
Let q > be a real number. (i) Let v := b ℓ +1 ( y, q ) . . . b ℓ + m ( y, q ) for some y ∈ [0 , and some integers ℓ ≥ and m ≥ . The set Y v of numbers x ∈ J q for which the word v does notoccur in the greedy expansion of x in base q has Hausdorff dimension lessthan one. (ii) The set Y of numbers x ∈ J q for which at least one word of the form b ℓ +1 ( y, q ) . . . b ℓ + m ( y, q ) ( ℓ ≥ , m ≥ , y ∈ [0 , does not occur in thegreedy expansion of x in base q is of first category, has Lebesgue measurezero and Hausdorff dimension one.Proof. (i) Using the inequality ( b i ( y, q )) < ( α i ( q )), it follows from Proposition 2.4that for some k ∈ N , there exist positive integers m , . . . , m k and nonnegativeintegers ℓ , . . . , ℓ k satisfying α m j ( q ) > ℓ j < α m j ( q ) for each 1 ≤ j ≤ k , suchthat v is a subword of w := α ( q ) . . . α m − ( q ) ℓ . . . α ( q ) . . . α m k − ( q ) ℓ k . Let W q and F ′ q be the same as the sets W r and F ′ r defined in the proof of theprevious theorem, but now with ( α i ) := ( α i ( q )) and n ≥ max { m , . . . , m k } largeenough such that the inequality(2.4) (cid:18) q n (cid:19) k < q m + ··· + m k holds. If w i j , . . . , w i k j k are k words belonging to W q such that i j . . . i k j k = ℓ m . . . ℓ k m k , we associate with them a similarity S i j ...i k j k : J q → J q defined by the formula S i j ...i k j k ( x ) = α q + · · · + α j − q j − + i q j + α q j +1 + · · · + α j − q j + j − + i q j + j ...+ α q j + ··· + j k − +1 + · · · + α j k − q j + ··· + j k − + i k q j + ··· + j k + xq j + ··· + j k , x ∈ J q . TWO-DIMENSIONAL UNIVOQUE SET 7
Let G ′ q denote the set of those sequences belonging to F ′ q which do not contain theword w , and let G q := ( ∞ X i =1 c i q i : ( c i ) ∈ G ′ q ) . Since ( α i ) = ( α i ( q )), a sequence belonging to F ′ q is not necessarily the greedyexpansion in base q of a number x ∈ J q , but this does not affect our proof. It isimportant, however, that any greedy expansion ( b i ) = α ∞ in base q can be writtenas α ℓ c c . . . for some ℓ ≥ c i ) belonging to F ′ q . If Y w denotesthe set of numbers x ∈ J q for which the word w does not occur in ( b i ( x, q )) thenthe latter fact implies that the set Y w \ { α / ( q − } can be covered by countablymany sets similar to G q .It follows from the definition of G q that G q ⊂ [ S i j ...i k j k ( G q )where the union runs over all i j . . . i k j k for which the similarity S i j ...i k j k isdefined above. Hence G q ⊂ [ S i j ...i k j k ( G q )and thus G q ⊂ H q where H q is the (nonempty compact) invariant set of this systemof similarities. Let ˜ α i := α i for 1 ≤ i < n and ˜ α n := α n + 1. From Proposition 9.6in [9] we know that dim H H q ≤ s where s is the real solution of the equation(2.5) n X j =1 n X j =1 · · · n X j k =1 (cid:18) Π ki =1 ˜ α j i q ( j + ··· + j k ) s (cid:19) − q ( m + ··· + m k ) s = 1 . Denoting the left side of (2.5) by C ( s ), we have C (1) + 1 q m + ··· + m k = n X i =1 ˜ α i q i ! k < (cid:18) q n (cid:19) k . By (2.4) we have C (1) <
1, and thus dim H Y v ≤ dim H Y w ≤ dim H H q < Y v ⊂ Y w ⊂ ∞ [ n =1 ( c n + d n H q )for some constants c n , d n ∈ R ( n ∈ N ). Arguing as in the proof of Theorem 2.7(ii)we may conclude that Y v is a null set of first category. Since Y is a countableunion of sets of the form Y v the same properties hold for the set Y . Let r ∈ (1 , q )and let G r,q be the set defined in Theorem 2.7. Due to Theorem 2.7(i) it is nowsufficient to show that G r,q ⊂ Y . By Proposition 2.1 there exists an integer n ∈ N such that the inequality α ( r ) . . . α n ( r ) < α ( q ) . . . α n ( q ). Note that the greedyexpansion in base q of a number x ∈ G r,q equals ( b i ( x ′ , r )) for some x ′ ∈ J r byProposition 2.4. Applying Propositions 2.1 and 2.4 once more we conclude thatthe sequence 0 α ( q ) . . . α n ( q )0 ∞ equals ( b i ( y, q )) for some y ∈ [0 ,
1) while the word0 α ( q ) . . . α n ( q ) does not occur in the greedy expansion in base r of any numberbelonging to J r . (cid:3) Remark.
In this remark we will briefly sketch a proof of Theorem 2.7(i) and The-orem 2.8(i) that was pointed out to us by the anonymous referee. For q >
1, let B n ( q ) be the number of possible blocks of length n that may occur in ( b i ( x, q )) forsome x ∈ J q . Since b n +1 ( x, q ) b n +2 ( x, q ) . . . is the greedy expansion of P ∞ i =1 b n + i q − i for each n ∈ N and x ∈ J q , we have B n ( q ) = | { ( b ( x, q ) , . . . , b n ( x, q )) : x ∈ J q } | . MARTIJN DE VRIES AND VILMOS KOMORNIK
Let the q - shift σ q be the one-sided left shift on the set { ( b i ( x, q )) : x ∈ J q } . It iswell known (see [12]) that the topological entropy h top ( σ q ) of the q -shift, given by(2.6) h top ( σ q ) := lim n →∞ log( B n ( q )) n , equals log q . By some modifications of the proof of Proposition III.1 in [10], oneshows that dim H G r,q = h top ( σ r ) / log q = log r/ log q . Theorem 2.8(i) may also bededuced from (2.6) and Proposition III.1 in [10]. On the other hand, our proof ofthese results enables us to show that the sets G q and Y in Theorem 2.7(ii) andTheorem 2.8(ii) are of first category. Moreover, Theorem 2.7(i) combined withthe formula dim H G r,q = h top ( σ r ) / log q gives an alternative proof of the fact that h top ( σ q ) = log q for each q > Proof of Theorem 1.2
The following characterization of unique expansions readily follows from Propo-sition 2.4.
Proposition 3.1.
Fix q > . A sequence ( c i ) of integers c i ∈ A q is the uniqueexpansion of some x ∈ J q if and only if c n +1 c n +2 . . . < α ( q ) α ( q ) . . . whenever c n < α ( q ) and c n +1 c n +2 . . . < α ( q ) α ( q ) . . . whenever c n > . In what follows we use the notation ( a i ( x, q )), ( b i ( x, q )), ( α i ( q )) and ( β i ( q )) asintroduced in Section 1. If x and q are clear from the context, then we omit thesearguments and we simply write a i , b i , α i and β i . If two couples ( x, q ) and ( x ′ , q ′ ) areconsidered simultaneously, then we also write a ′ i , b ′ i , α ′ i and β ′ i in place of a i ( x ′ , q ′ ), b i ( x ′ , q ′ ), α i ( q ′ ) and β i ( q ′ ). Lemma 3.2.
Given ( x, q ) ∈ J , the following two conditions are equivalent: a n +1 a n +2 . . . ≤ α α . . . whenever a n > a n +1 a n +2 . . . ≤ β β . . . whenever a n > . Proof.
Since ( α i ) ≤ ( β i ), it suffices to show that if there exists a positive integer n such that a n > a n +1 a n +2 . . . > α α . . . , then there exists also a positive integer m such that a m > a m +1 a m +2 . . . > β β . . . . If the greedy expansion ( β i ) is infinite, then ( β i ) = ( α i ) and we may choose m = n .If ( β i ) has a last nonzero digit β ℓ , then ( α i ) = ( α . . . α ℓ ) ∞ with α . . . α ℓ − α ℓ = β . . . β ℓ − β − ℓ ( β − ℓ := β ℓ − α ℓ < α . Since we have a n +1 a n +2 . . . > ( α . . . α ℓ ) ∞ by assumption, there exists a nonnegative integer j satisfying a n +1 . . . a n + jℓ = ( α . . . α ℓ ) j and a n + jℓ +1 . . . a n +( j +1) ℓ > α . . . α ℓ . Putting m := n + jℓ it follows that a m > a m +1 . . . a m + ℓ ≥ β . . . β ℓ . It follows from our assumption a n +1 a n +2 . . . > α α . . . that ( α i ) < α ∞ and ( a i ) = α ∞ . It follows from Proposition 2.2 that ( a i ) has no tail equal to α ∞ , so that a m + ℓ +1 a m + ℓ +2 . . . > ∞ . We conclude that a m +1 a m +2 . . . > β β . . . . (cid:3) TWO-DIMENSIONAL UNIVOQUE SET 9
Definition.
We say that ( x, q ) ∈ J belongs to the set V if one of the two equivalentconditions of the preceding lemma is satisfied. Moreover, we define V q := { x ∈ J q : ( x, q ) ∈ V } , q > . It follows from Proposition 3.1 that U ⊂ V ⊂ J . Proof of Theorem 1.2.
We need to prove that U ∩ J = V .First we show that V ⊂ U . To this end we introduce for each fixed q > U ′ q and V ′ q , defined by U ′ q := { ( a i ( x, q )) : x ∈ U q } and V ′ q := { ( a i ( x, q )) : x ∈ V q } . Observe that U ′ q is simply the set of unique expansions in base q . It follows easilyfrom Propositions 2.1, 2.2 and 3.1 that U ′ q ⊂ V ′ q for each q >
1, and that V ′ q ⊂ U ′ r for each r > q such that ⌈ q ⌉ = ⌈ r ⌉ . Since we also have U q = V q = [0 ,
1] if q > U ∩ J ⊂ V . Since U ⊂ V it is sufficient to prove that if( x, q ) ∈ J \ V , then ( x ′ , q ′ ) / ∈ V for all ( x ′ , q ′ ) ∈ J close enough to ( x, q ). By Lemma3.2 there exist two positive integers n and m such that(3.1) a n > a n +1 . . . a n + m > β . . . β m . This implies in particular that q is not an integer, because otherwise ( α i ) = ( β i ) = β ∞ . Hence, if q ′ is sufficiently close to q , then(3.2) β ′ . . . β ′ m ≤ β . . . β m by Lemma 2.5. It follows from the definition of quasi-greedy expansions that a q + · · · + a j − q j − + a + j q j + 1 q j + m > x whenever a j < α ,where a + j := a j + 1. If ( x ′ , q ′ ) ∈ J is sufficiently close to ( x, q ), then α = α ′ , theinequality (3.2) is satisfied, a ′ . . . a ′ n + m ≥ a . . . a n + m by Lemma 2.3, and(3.3) a q ′ + · · · + a j − ( q ′ ) j − + a + j ( q ′ ) j + 1( q ′ ) j + m > x ′ whenever j ≤ n + m and a j < α .Now we distinguish between two cases.If a ′ . . . a ′ n + m = a . . . a n + m , then we have a ′ n > a ′ n +1 . . . a ′ n + m > β . . . β m ≥ β ′ . . . β ′ m by (3.1) and (3.2). This proves that ( x ′ , q ′ ) / ∈ V .If a ′ . . . a ′ n + m > a . . . a n + m , then let us consider the smallest j for which a ′ j > a j .It follows from (3.2) and (3.3) that a ′ j = a + j > a ′ j +1 . . . a ′ j + m = β m > β . . . β m ≥ β ′ . . . β ′ m . Hence ( x ′ , q ′ ) / ∈ V again. (cid:3) Remark.
It is the purpose of this remark to describe the set U \ J . For each m ∈ N ,we define the number q m ∈ ( m, m + 1) by the equation1 = mq m + 1 q m . Fix q ∈ ( m, q m ]. Since α ( q ) = m and α ( q ) = 0, Proposition 3.1 implies that asequence ( c i ) ∈ { , . . . , m } N belongs to U ′ q if and only if for each n ∈ N , we have c n < m = ⇒ c n +1 < m and c n > ⇒ c n +1 > . Denoting the set of all such sequences by D ′ m and putting for m > D ′ = { ∞ , ∞ } ), D m := ( ∞ X i =1 c i m i : ( c i ) ∈ D ′ m ) , one may verify that U \ J = { (0 , } ∪ ∞ [ m =2 ( D m \ [0 , × { m } . For x ≥
0, let U ( x ) = { q > x, q ) ∈ U } , and denote its closure by U ( x ). Usingthis notation, the set U introduced in Section 1 equals U (1). The following corollaryimplies in particular that the sets U ( x ) \ U ( x ) are (at most) countable. Corollary 3.3.
Each element q ∈ U ( x ) \ U ( x ) is algebraic over the field Q ( x ) .Proof. If q ∈ U ( x ) \ U ( x ) and q / ∈ N , then ( x, q ) ∈ J and thus ( x, q ) ∈ V byTheorem 1.2. If the sequence ( b i ( x, q )) is infinite, then it ends with α ( q ) α ( q ) . . . as follows from the definition of V and Propositions 2.4 and 3.1. Hence x has afinite expansion in base q or x can be written as x = b ( x, q ) q + · · · + b n ( x, q ) q n + 1 q n (cid:18) α q − − (cid:19) for some n ≥
0, whence q is algebraic over Q ( x ). (cid:3) Proof of Theorem 1.1
We need some results on the Hausdorff dimension of the sets U q and V q for q >
1. It follows from Theorem 1.2 that U q ⊂ U q ⊂ V q . Moreover, if an element x ∈ V q \ U q has an infinite greedy expansion in base q , then ( b i ( x, q )) must end with α ( q ) α ( q ) . . . as follows from Propositions 2.4 and 3.1; hence V q \ U q is (at most)countable and the sets U q , U q and V q have the same Hausdorff dimension for each q >
1. Proposition 4.1 below is contained in the works of Dar´oczy and K´atai [4],Kall´os [13], [14], Glendinning and Sidorov [11], and Sidorov [21]; for the reader’sconvenience we provide here an elementary proof.
Proposition 4.1.
We have (i) lim q ↑ dim H U q = 1 ; (ii) dim H U q < for all non-integer q > .Proof. (i) Assume that q ∈ (1 ,
2) is larger than the tribonacci number, i.e.,1 q + 1 q + 1 q < , and let N = N ( q ) ≥ q + · · · + 1 q N − < . Hence α ( q ) = · · · = α N − ( q ) = 1. Let us denote by I q the set of numbers x ∈ J q which have an expansion ( c i ) in base q satisfying 0 < c kN +1 + · · · + c ( k +1) N < N forevery nonnegative integer k . Since in such expansions ( c i ), a zero (one) is followedby at most 2 N − I q ⊂ U q . It suffices now to prove that(4.1) dim H I q = log(2 N − N log q ;indeed, q ↑ N → ∞ , hence dim H I q → H U q → TWO-DIMENSIONAL UNIVOQUE SET 11
Observe that(4.2) I q = [ S c ...c N ( I q )where the union runs over the words c . . . c N of length N consisting of zeros andones satisfying 0 < c + · · · + c N < N , and S c ...c N : J q → J q is given by S c ...c N ( x ) := (cid:16) c q + · · · + c N q N (cid:17) + xq N , x ∈ J q . Moreover, the set I q is closed (and thus compact) because the limit of a monotonicsequence in I q converges to an element of I q . In other words, I q is the (nonemptycompact) invariant set of the iterated function system formed by these 2 N − S c ...c N ( I q ) on the right side of (4.2) are disjoint because S c ...c N ( I q ) ⊂ I q ⊂ U q , and since all similarity ratios are equal to q − N , it followsfrom Propositions 9.6 and 9.7 in [9] that the Hausdorff dimension s of I q is the realsolution of the equation (2 N − q − Ns = 1 , which is equivalent to (4.1).(ii) Let q > n ∈ N be such that α n ( q ) < α ( q ). Itfollows from Proposition 3.1 that the word 1(0) n does not occur in ( b i ( x, q )) if x belongs to U q . Applying Theorem 2.8(i) with y = q − , ℓ = 0 and m = n + 1, weconclude that dim H U q < (cid:3) Proof of Theorem 1.1. (ii) Let q > V q \ U q is countable,Proposition 4.1(ii) yields that dim H V q <
1. This implies in particular that the set V q is a one-dimensional null set. Applying Theorem 1.2 (and the remark followingits proof) and Fubini’s theorem we conclude that U is a two-dimensional null set.(i) Since U q is not closed for all q > U cannot be closed. Since U is a two-dimensional null set, it has no interior points. It remains to show that U (and thus U ) has no isolated points. If q > U q is densein J q = [0 , q > x, q ) ∈ U is not isolated because U ′ q ⊂ U ′ r whenever q < r and ⌈ q ⌉ = ⌈ r ⌉ .(iii) From Corollary 7.10 in [9] we may conclude that for almost all q > H U q ≤ max { , dim H U − } which would contradict Proposition 4.1(i) if we had dim H U < (cid:3) Acknowledgements.
We warmly thank the anonymous referee for suggesting alter-native proofs of Theorem 2.7(i) and Theorem 2.8(i) (see the last remark of Sec-tion 2), and for a very careful reading of the manuscript. The first author has beensupported by NWO, Project nr. ISK04G. Part of this work was done during avisit of the second author at the Department of Mathematics of the Delft Univer-sity of Technology. He is grateful for this invitation and for the excellent workingconditions.
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