AA number system with base − ∗ and J¨org M. ThuswaldnerMontanuniversit¨at Leoben, Franz Josef Straße 18, A-8700 Leoben, Austria Abstract
In the present paper we explore a way to represent numbers with respect to the base − using the set of digits { , , } . Although this number system shares several propertieswith the classical decimal system, it shows remarkable differences and reveals interesting newfeatures. For instance, it is related to the field of 2-adic numbers, and to some “fractal” setthat gives rise to a tiling of a non-Euclidean space. A number system is, intuitively, a way of representing a certain set of numbers in a consistentmanner, using strings of some given digits in relation to a base. The most famous examples arethe decimal and the binary systems. Over time, many generalizations of these number systemscame to the fore. They have applications in various areas of mathematics and computer science.To cite some examples, back in 1885 Gr¨unwald [5] studied number systems with negative integersas bases. In 1936 Kempner [8] and later also R´enyi [12] proposed expansions w.r.t. nonintegralreal bases (see e.g. also [14]). Knuth [9] introduced complex bases and dealt with their relationsto fractal sets (see [10, p. 608] for an important example). Gordon [4] discussed the relevanceof number systems with varying digit sets in cryptography and in [11] binary and hexadecimalexpansions were used in this context.Figure 1: The tile F related to the number system with base − .The aim of this article is to define and explore a number system which has base − and usesthe set of digits { , , } . We chose this example because it has many beautiful properties thatlink different areas of mathematics, and nevertheless can be studied in a way that is accessible to ∗ The doctoral position of this author is supported by the Austrian Science Fund (FWF), project W1230. a r X i v : . [ m a t h . N T ] F e b broad readership. The results stated here are known in a more general context; however, sinceour proofs are presented in terms of our particular example, we can avoid the use of advancedtechniques. Number systems of a very similar kind were introduced by Akiyama et al. [1] andgeneralizations of this have been studied in [3] and [13]. We mention that there are other ways todefine a number system with base − , see for example the one using the digit set { , } in [2].Proceeding in analogy to the decimal system, we first investigate expansions of integers and realnumbers. Realizing that the real line is somehow “too small” for our number system, we introducethe space K = R × Q , where Q is the field of 2-adic numbers, and show how it naturally arisesas a representation space. We relate the set F depicted in Figure 1 to our number system. Thisset has a self-affine structure and many nice properties. Among other things, we prove that F induces a tiling of K by translations, and relate this tiling to the existence and uniqueness ofexpansions w.r.t. the base − in K . The set F is an example of a rational self-affine tile . Steinerand Thuswaldner [16] dealt with such tiles in a much more general framework. In the decimal system, each integer can be expanded without using digits after the decimal point.In this section, we wish to define and explore such “integer expansions” in the number systemwith base − and digit set D = { , , } , which we denote as ( − , D ). The fact that − is not aninteger will entail new properties. The reason for the choice of a negative base is that there willbe no need for a minus sign to represent negative numbers.To get a feeling for this number system, we first deal with the integers and follow the ideas of[1]. Given N ∈ Z , we want to find an expansion of the form N = k (cid:88) i =0 d i (cid:0) − (cid:1) i ( k ∈ N , d i ∈ D ) . (1)From here onwards, we assume d k (cid:54) = 0 whenever k (cid:62)
1. The factor at the beginning is justthere for convenience (and to be consistent with [1]) and will not be crucial. In order to producean expansion of the form (1), we use the following algorithm. Write2 N = − N + d with d ∈ D , N ∈ Z . Since N is given, d has to be the unique digit satisfying d ≡ N (mod 3),so this equation has a unique solution N . More generally, we set N = N and recursively definethe integers N i +1 for i (cid:62) N i = − N i +1 + d i (2)with d i ∈ D . One can easily see by induction on i that this yields N = (cid:0) − (cid:1) i +1 N i +1 + d i (cid:0) − (cid:1) i + · · · + d (cid:0) − (cid:1) + d . (3)If we can prove that N k = 0 for k large enough, our algorithm gives the desired representation (1)for each N ∈ Z . And this is indeed our first result. Proposition 1.
Each N ∈ Z can be represented in the form (1) in a unique way.Proof. By (3), it suffices to show that for each N = N ∈ Z the sequence ( N i ) i (cid:62) producedby the recurrence (2) is eventually zero. We have N i +1 = − N i + d i with d i ∈ D , hence | N i +1 | (cid:54) | N i | + and therefore | N i +1 | < | N i | holds for each | N i | (cid:62)
3. This implies that there is i ∈ N with | N i | (cid:54) N i = − N i +1 = 2, N i +2 = − N i +3 = 1, N i +4 = 0, and N i +5 = 0. Thus for each N = N ∈ Z there is k ∈ N with N k = 0 for all k (cid:62) k and N has anexpansion of the form (1).Concerning uniqueness of the expansion, we just note that each digit d i in (1) has to lie in aprescribed residue class modulo 3, and hence is uniquely determined.2e write ( d k . . . d ) − / := k (cid:88) i =0 d i (cid:0) − (cid:1) i ( k ∈ N , d i ∈ D )and call ( d k . . . d ) − / an integer ( − )-expansion . We proved in Proposition 1 that each N ∈ Z has a unique integer ( − )-expansion. For instance, − − / and 4 = (21122) − / .As a next step we characterize the set D [ − ] = (cid:8) ( d k . . . d ) − / | k ∈ N , d i ∈ D (cid:9) of all real numbers with an integer ( − )-expansion (see also [13, Example 3.3]). Theorem 2.
The set of all numbers having an integer ( − ) -expansion is Z [ ] . Here, as usual,we set Z [ ] = { a − (cid:96) | a ∈ Z , (cid:96) ∈ N } .Proof. We have to show that D [ − ] = Z [ ] . The inclusion D [ − ] ⊂ Z [ ] is trivial. For thereverse inclusion, let N ∈ Z [ ] be arbitrary. There exist a ∈ Z and (cid:96) ∈ N such that N = a − (cid:96) .We need to show that N has an expansion of the form (1) for some k ∈ N . We define the samerecurrence as in (2) but forcing N i ∈ Z [ ] for i (cid:62)
0, and proceed to show that N k = 0 for some k ∈ N .Since N is given by 2 N = − N + d , where N ∈ Z [ ] and d ∈ D , we have − N = 2 N − d = a (cid:96) − − d = a − (cid:96) − d (cid:96) − . (4)To guarantee that N ∈ Z [ ], the numerator of this fraction has to be divisible by 3, namely, weneed to choose d in a way that 2 (cid:96) − d ≡ a (mod 3). As the inverse of 2 (cid:96) − in Z / Z is 2 (cid:96) − , weget d ≡ (cid:96) − a (mod 3) and N is uniquely defined by (4).Iterating (2) yields that N i = a i − (cid:96) + i for some a i ∈ Z for i ∈ { , . . . , (cid:96) } . After (cid:96) steps, we get N (cid:96) ∈ Z , and we are in the case covered by Proposition 1. This implies that there is k ∈ N suchthat N k = 0 for k (cid:62) k and thus N ∈ D [ − ].By residue class considerations one can show that each z ∈ Z [ ] has a unique integer ( − )-expansion. For instance, − = (120) − / and = (111) − / . We proceed to study ( − )-expansions for arbitrary reals, allowing negative powers of the base.We will consider the matter of uniqueness and motivate the future construction of a tiling.A desirable property of a number system is that almost all numbers (in a measure theoreticsense) can be expanded in a unique way. For example, although the decimal expansion is notalways unique ( e.g. the number 4 can also be written as 3 . ... ), the set of numbers admittingmore than one decimal expansion is very small in the sense that it has Lebesgue measure zero.This fact is reflected by the following tiling property . Consider the set of fractional parts in thedecimal number system, that is, the set of numbers that can be expanded using only negativepowers of 10: it corresponds to the unit interval [0 , integer expansion ,is equal to Z (if we permit the use of the minus sign). Consider the collection { [0 ,
1] + z | z ∈ Z } . (5)This collection covers R , and the only overlaps occur in the boundary points of the intervals,which form a measure zero set. We thus say that { [0 ,
1] + z | z ∈ Z } forms a tiling of the realline. Here [0 ,
1] is the central tile and Z is the translation set . This tiling property is a geometricinterpretation of the fact that almost all real numbers admit a unique expansion in the decimal3ystem, and it works the same for any other q -ary number system ( q ∈ Z ; | q | (cid:62) − , D ) is a priori neverunique. Let ( d k . . . d .d − d − . . . ) − / := k (cid:88) i = −∞ d i (cid:0) − (cid:1) i ( k ∈ Z , d i ∈ D ) (6)and consider Ω = { (0 .d − d − . . . ) − / | d − i ∈ D} , (7)the set of fractional parts in ( − , D ). One can prove that Ω = (cid:2) − , (cid:3) by using the fact that − Ω = Ω ∪ (cid:0) Ω + (cid:1) ∪ ( Ω + 1) (we revisit this idea later on in Section 5).Each decomposition of a real number x as the sum of an element of Z [ ] and an element of Ω leads to an expansion of x in the form (6), by Theorem 2 and the definition of Ω . Because thecollection { Ω + z | z ∈ Z [ ] } covers the real line, each real number can be written as such a sum,and hence admits an expansion of the form (6). But since each x ∈ R is contained in multipleelements of the collection { Ω + z | z ∈ Z [ ] } (in fact, in infinitely many), it admits multipleexpansions of the form (6). For example, = (0 . . . . ) − / = (2 . . . . ) − / .Different translations of Ω by elements of Z [ ] overlap in sets of positive measure; in otherwords, we do not have the desired tiling property. This results in expansions that are not unique.In the subsequent sections, we will find a way to “embed” the collection { Ω + z | z ∈ Z [ ] } in asuitable space where it will give rise to a tiling. The real line seems to be “too small” for the collection { (cid:2) − , (cid:3) + z | z ∈ Z [ ] } , so we wishto enlarge the space R in order to mend the issue with the overlaps . Indeed, our next goal is todefine a new space, called K , in which the number system ( − , D ) induces a tiling in a naturalway. The idea behind this is as follows: the overlaps occur because the three digits { , , } are“too many” for a base whose modulus is . Such a base would need one and a half digits, which isof course not doable. What causes all the problems is the denominator 2. Roughly speaking, thisdenominator piles up powers of two which are responsible for the overlaps. It turns out that theseoverlaps can be “unfolded” by adding a 2 -adic factor to our representation space. The strategyof enlarging the representation space by p -adic factors that we are about to present was usedin the setting of substitution dynamical systems e.g. by Siegel [15] and in a much more generalframework than ours in [16].We begin by introducing the 2-adic numbers; for more on this topic we refer the reader to [6].Consider a nonzero rational number y and write y = 2 (cid:96) pq where (cid:96) ∈ Z and both p and q are odd.The 2-adic absolute value in Q is defined by | y | = (cid:40) − (cid:96) , if y (cid:54) = 0 , , if y = 0 , and the 2-adic distance between two rationals x and y is given by | x − y | . Two points are closeunder this metric if their difference is divisible by a large positive power of 2.We define Q to be the completion of Q with respect to | · | . The space Q is a field calledthe field of -adic numbers . Every nonzero y ∈ Q can be written uniquely as a series y = ∞ (cid:88) i = (cid:96) c i i ( (cid:96) ∈ Z , c i ∈ { , } , c (cid:96) (cid:54) = 0) . This series converges in Q because large powers of two have small 2-adic absolute value. Indeed,we have | y | = 2 − (cid:96) . Another way, which we do not pursue here, would be to restrict the “admissible” digit strings, see [1].
4e define our representation space as K = R × Q , with the additive group structure given bycomponentwise addition. Moreover, Z [ ] acts on K by multiplication, more precisely, if α ∈ Z [ ]and ( x , x ) ∈ K then α · ( x , x ) = ( αx , αx ) = ( x , x ) · α. For every ( x , x ) , ( y , y ) ∈ K define d (( x , x ) , ( y , y )) := max {| x − y | , | x − y | } . Then d is a metric on K . Intuitively, two points in K are far apart if either their real componentsare far apart or their 2-adic components are far apart.We define the embedding ϕ : Q → K , z (cid:55)→ ( z, z ) . Consider the image of Z [ ] under ϕ . Despite both coordinates of ϕ ( z ) being the same, it does notlie in a diagonal. Indeed, points of Z [ ] that are close in R are far apart in the 2-adic distance. Inparticular, we will show that the points of ϕ ( Z [ ]) form a lattice .A subset Λ of K is a lattice if it satisfies the three following conditions.1. Λ is a group.2. Λ is uniformly discrete, meaning there exists r > r in K contains at most one point of Λ.3. Λ is relatively dense, meaning there exists R > R in K contains at least one point of Λ. Lemma 3. ϕ ( Z [ ]) is a lattice in K .Proof.
1. The fact that ϕ ( Z [ ]) is a group follows from the additive group structure of Z [ ]because ϕ is a group homomorphism.2. To get uniform discreteness of ϕ ( Z [ ]) we show first that d ( ϕ ( z ) , ϕ (0)) (cid:62) z ∈ Z [ ]. Recall that d ( ϕ ( z ) , ϕ (0)) = max {| z | , | z | } . If | z | < a, (cid:96) ∈ Z with a odd and (cid:96) ≥ z = a − (cid:96) , so | z | = 2 (cid:96) > d ( ϕ ( z ) , ϕ (0)) (cid:62) ϕ ( Z [ ]) is at least one, hence ϕ ( Z [ ]) is uniformly discrete.3. For the relative denseness, consider an arbitrary element ( x , x ) ∈ K . We claim that thereexists z ∈ Z [ ] such that d ( ϕ ( z ) , ( x , x )) (cid:54)
2. Let z ∈ Z be one of the integers beingclosest to x . If x = (cid:80) ∞ i = (cid:96) c i i with (cid:96) ∈ Z and c i ∈ { , } then set z = (cid:80) − i = (cid:96) c i i ∈ Z [ ](note that z = 0 if (cid:96) (cid:62) d (( x , x ) , ( z , z )) = max {| x − z | , | x − z | } (cid:54) . Now we set z = z + z ∈ Z [ ]. Because z is an integer, | z | (cid:54)
1, and since z ∈ [0 , | z | (cid:54)
1. Thus d ( ϕ ( z ) , ( z , z )) = max {| z | , | z | } (cid:54)
1, and so d ( ϕ ( z ) , ( x , x )) (cid:54) ϕ ( Z [ ]) is relatively dense.Figure 2 illustrates some points of ϕ ( Z [ ]). Drawing pictures in this setting is not straightfor-ward: the space K is non-Euclidean, so we need to represent it in R while somehow maintainingthe 2-adic nature of the second component. We do this in the following way: any point y ∈ Q canbe written uniquely (up to “leading zeros”) as a series y = (cid:80) ∞ i = (cid:96) c i ( − ) i with (cid:96) ∈ Z , c i ∈ { , } ,that converges in the 2-adic metric (this is a 2-adic expansion, not a ( − )-expansion!). We considerthe mapping γ : Q → R ; ∞ (cid:88) i = (cid:96) c i ( − ) i (cid:55)→ ∞ (cid:88) i = (cid:96) c i − i , (8)5igure 2: Representation of the lattice points ϕ ( j ) for − (cid:54) j (cid:54)
64 in R .which is well defined since the sum on the right hand side converges in R . A point ( x , x ) ∈ K isnow represented as ( x , γ ( x )) ∈ R .The lattice ϕ ( Z [ ]), which will play the role of the “integers” in K , turns out to be a propertranslation set for a tiling of K related to the number system ( − , D ). In the real case, whendefining a tiling we allowed overlaps as long as it was on a set of Lebesgue measure zero. In orderto generalize this, we need to define a natural measure on K .A Haar measure is a translation invariant Borel measure that is finite for compact sets. It canbe defined in spaces with a sufficiently “nice” structure (more specifically, it is defined on locallycompact topological groups). Such a measure is unique up to a scaling factor. The Lebesguemeasure µ ∞ in R is a Haar measure.Let µ be the Haar measure in Q that satisfies µ (2 (cid:96) Z ) = 2 − (cid:96) , where Z = (cid:8) (cid:80) ∞ i =0 c i i | c i ∈ { , } (cid:9) ⊂ Q is the ring of 2-adic integers. This is a very naturalmeasure: multiplying by large powers of two makes a set small in measure.Let µ = µ ∞ × µ be the product measure of µ ∞ and µ on K = R × Q , that is, if M ⊂ R and M ⊂ Q are respectively measurable, then the sets of the form M = M × M generate the σ -algebra of µ , and µ ( M ) := µ ∞ ( M ) µ ( M ). One can show that µ is a Haar measure on K .For any measurable set M = M × M ⊂ K , we have µ ∞ ( − M ) = µ ∞ ( M ) , µ ( − M ) = 2 µ ( M )which yields µ ( − M ) = µ ∞ ( − M ) µ ( − M ) = 3 µ ( M ) . (9)Thus multiplying any measurable set M ⊂ K by the base − enlarges the measure by the factor3, which can be interpreted as having “enough space” for three digits. F . In this section we define a set
F ⊂ K that plays the same role for ( − , D ) as the unit intervaldoes for the decimal system. We explore some of its topological and measure theoretic properties.Recall that in (7) we defined the set Ω of fractional parts, consisting of elements of the form(0 .d − d − . . . ) − / . We now embed the digits in K , obtaining the set F := (cid:110) ∞ (cid:88) i =1 ϕ ( d − i ) (cid:0) − (cid:1) − i | d − i ∈ D (cid:111) . F is a compact subset of K . Indeed, given any sequence in F , we use a Cantor diagonalargument to find a convergent subsequence.Let x ∈ F : if we multiply x by the base − , we obtain − x ∈ F + ϕ ( d − ) with d − ∈ { , , } (this can be interpreted as the analog of moving the decimal point one place to the right). Thus F satisfies the set equation − F = F ∪ (cid:0) F + ϕ ( ) (cid:1) ∪ (cid:0) F + ϕ (1) (cid:1) (10)in K , which can be written shortly as − F = F + ϕ ( D ) . It turns out that this set equationcompletely characterizes F . Note that (10) is equivalent to F = (cid:0) − (cid:1) F ∪ (cid:0) − (cid:1) (cid:0) F + ϕ ( ) (cid:1) ∪ (cid:0) − (cid:1) (cid:0) F + ϕ (1) (cid:1) , (11)and multiplying by − is a uniform contraction in K : it is a contraction in R because |− | < Q because |− | = < . Thus (11) states that F is equal to the union of three contractedcopies of itself. Because of this contraction property we may apply Hutchinson’s Theorem (see [7])which says that there exists a unique nonempty compact subset of K that satisfies the set equation(11). Thus F is uniquely defined as the nonempty compact set satisfying (11) (or, equivalently,(10)). The set F is called a rational self-affine tile in the sense of [16].Since according to Theorem 2 the set Z [ ] is the analog of Z in the number system ( − , D ),we define the analog of the collection in (5) by setting C = {F + ϕ ( z ) | z ∈ Z [ ] } . Then C is a collection of copies of F translated by elements of the lattice ϕ ( Z [ ]). We will showin Theorem 6 that C is a tiling of K , meaning that:1. C is a covering of K , i.e. , (cid:104)C(cid:105) = K , where (cid:104)C(cid:105) = F + ϕ ( Z [ ]) is the union of the elements of C .2. Almost every point in K (with respect to the measure µ ) is contained in exactly one elementof C .Figure 1 shows a representation of F in R , again using the function γ from (8) to map Q to R . Figure 3 shows a patch of C ; the translates of F appear to have different shapes, but this isFigure 3: A patch of the tiling C of K by translates of F .due to the embedding of K in R . The illustration indicates, however, that the different translatesof F do not overlap other than in their boundaries.Because F plays the role of the central tile, we require it to be a reasonably nice set topologicallyspeaking. As a next step, we prove that F is the closure of its interior and that its boundary ∂ F has measure zero. In a general setting, this result is contained in [16, Theorem 1].7 heorem 4. F is the closure of its interior.Proof. We first prove that C is a covering of K , i.e. , (cid:104)C(cid:105) = K , where (cid:104)C(cid:105) is the union of the elementsof C . Applying the set equation (10) we obtain − (cid:104)C(cid:105) = − F − ϕ ( Z [ ]) = F + ϕ ( D ) − ϕ ( Z [ ]) . Note that D = { , , } is a complete set of representatives of residue classes of Z [ ] / ( − ) Z [ ],so ϕ ( D ) − ϕ ( Z [ ]) = ϕ ( Z [ ]) . Thus − (cid:104)C(cid:105) = (cid:104)C(cid:105) and, a fortiori , for any k ∈ N we have ( − ) k (cid:104)C(cid:105) = (cid:104)C(cid:105) . Recall that multiplyingby − is a contraction in K . We have shown in Lemma 3 that ϕ ( Z [ ]) is a relatively dense set in K , and therefore so is (cid:104)C(cid:105) , meaning there is some R > R intersects (cid:104)C(cid:105) . But since (cid:104)C(cid:105) is invariant under contractions by ( − ) k , this implies that any ballof radius ( ) k R with k ∈ N intersects (cid:104)C(cid:105) , hence it is dense in K .Consider now an arbitrary point in x ∈ K and a bounded neighborhood V of x . Since ϕ ( Z [ ])is uniformly discrete and V is bounded, V intersects only a finite number of translates of F , eachof which is compact. Since (cid:104)C(cid:105) is dense in K , x cannot be at positive distance from all thesetranslates of F . Thus x is contained in some translate of F and, hence, x ∈ (cid:104)C(cid:105) . Since x wasarbitrary this implies that (cid:104)C(cid:105) = K .Next, we show that int F (cid:54) = ∅ . Assume on the contrary that int F = ∅ . Consider the sets U z := K \ ( F + ϕ ( z )) ( z ∈ Z [ ]) . By assumption, U z is dense in K for each z ∈ Z [ ], and { U z | z ∈ Z [ ] } is a countable collection.Baire’s theorem asserts that a countable intersection of dense sets is dense. But (cid:92) z ∈ Z [ ] U z = K \ (cid:91) z ∈ Z [ ] F + ϕ ( z ) = K \ (cid:104)C(cid:105) = ∅ , which is clearly not dense. This contradiction yields int F (cid:54) = ∅ .We now prove the result. Iterating the set equation (10) for k ∈ N times yields F = (cid:0) − (cid:1) k F + (cid:16) ϕ ( D ) (cid:0) − (cid:1) k + ϕ ( D ) (cid:0) − (cid:1) k − + · · · + ϕ ( D ) (cid:0) − (cid:1)(cid:17) . Setting D k := D + D (cid:0) − (cid:1) + · · · + D (cid:0) − (cid:1) k − (12)this becomes F = ( − ) k ( F + ϕ ( D k )) ( k ∈ N ) , (13)which means we can write F as a finite union of arbitrarily small shrunk translated copies of itself.We know that F has an inner point x , therefore each copy of the form ( − ) k ( F + ϕ ( d )) , d ∈ D k ,has an inner point. Thus for any y ∈ F and any ε > k ∈ N and d ∈ D k so thatdiam(( − ) k ( F + ϕ ( d ))) < ε and y ∈ ( − ) k ( F + ϕ ( d )). Thus there is an inner point at distanceless than ε from y . Since y ∈ F and ε > F is the closure of itsinterior. Theorem 5.
The boundary of F has measure zero.Proof. Let x be an inner point of F and B ε ( x ) ⊂ F an open ball of radius ε > x . Because multiplication by − is a uniform contraction in K , there is k ∈ N such thatdiam ( − ) k F < ε . Thus by (13) there is d ∈ D k such that (cid:0) − (cid:1) k (cid:0) F + ϕ ( d ) (cid:1) ⊂ B ε ( x ) ⊂ int F . y ∈ ∂ (cid:0) (cid:0) − (cid:1) k (cid:0) F + ϕ ( d ) (cid:1)(cid:1) ⊂ int F . Since y is also an inner point of F , and (13) writes F as a finite union of compact sets, y mustnecessarily lie in ( − ) k ( F + ϕ ( d )) for some d ∈ D k \ { d } . Thus the boundary ∂ (( − ) k ( F + ϕ ( d ))) is covered at least twice by the collection { ( − ) k ( F + ϕ ( d )) | d ∈ D k } . This entailsthat µ ( F ) = µ (cid:0) (cid:91) d ∈D k (cid:0) − (cid:1) k (cid:0) F + ϕ ( d ) (cid:1)(cid:1) (cid:54) (cid:88) d ∈D k µ (cid:0) (cid:0) − (cid:1) k (cid:0) F + ϕ ( d ) (cid:1)(cid:1) − µ (cid:0) ∂ (cid:0) (cid:0) − (cid:1) k (cid:0) F + ϕ ( d ) (cid:1)(cid:1)(cid:1) . Note that (as a Haar measure) µ is translation invariant, the cardinality of D k is 3 k , and from (9)it follows that µ (( − ) k F ) = 3 − k µ ( F ). All this combined yields µ ( F ) (cid:54) (cid:88) d ∈D k µ (cid:0) (cid:0) − (cid:1) k F (cid:1) − µ (cid:0) ∂ (cid:0) (cid:0) − (cid:1) k F (cid:1)(cid:1) (cid:54) k − k µ ( F ) − µ (cid:0) ∂ (cid:0) (cid:0) − (cid:1) k F (cid:1)(cid:1) = µ ( F ) − µ (cid:0) ∂ (cid:0) (cid:0) − (cid:1) k F (cid:1)(cid:1) and therefore µ ( ∂ (( − ) k F )) = 0. This implies that µ ( ∂ F ) = 0. This section contains our final result: a tiling theorem for the ( − )-number system. This resultis contained in [16, Theorem 2] in a more general setting. For our special case, the proof is muchsimpler. As mentioned before, the tiling property is important because it relates to the uniqueness(almost everywhere) of expansions in the ( − )-number system embedded in K . We prove this asa corollary of our tiling theorem. Theorem 6.
The collection C = {F + ϕ ( z ) | z ∈ Z [ ] } forms a tiling of K .Proof. We have shown in the proof of Theorem 4 that C is a covering of K . It remains to showthat almost every point of K is covered by exactly one element of the collection C . Recall thatfor each k (cid:62)
1, the sets D k (see (12)) consist of all the integer ( − )-expansions with at most k digits. According to Theorem 2, the set Z [ ] is the set of all integer ( − )-expansions. Thisimplies that Z [ ] = (cid:83) k (cid:62) D k and, hence, K = F + ϕ ( Z [ ]) = (cid:83) k (cid:62) F + ϕ ( D k ) . Therefore, itsuffices to prove that the collection {F + ϕ ( d ) | d ∈ D k } has essentially disjoint elements foreach k (cid:62)
1, that is, if d, d (cid:48) ∈ D k are distinct then µ (( F + ϕ ( d )) ∩ ( F + ϕ ( d (cid:48) ))) = 0. Applying(13) we obtain3 k µ ( F ) = µ (( − ) k F ) = µ ( (cid:91) d ∈D k F + ϕ ( d )) (cid:54) (cid:88) d ∈D k µ ( F + ϕ ( d )) = 3 k µ ( F ) . This implies equality everywhere and, hence, different ϕ ( D k )-translates of F only overlap in setsof measure zero. Thus the same is true for different Z [ ]-translates of F . So the tiles in C areessentially disjoint, and C is a tiling. Corollary 7.
Almost every point x ∈ K has a unique expansion of the form x = k (cid:88) i = −∞ (cid:0) − (cid:1) i ϕ ( d i ) ( k ∈ N , d i ∈ D ; d k (cid:54) = 0 whenever k (cid:62) . (14)9 roof. Let x ∈ K and suppose it has two different expansions x = k (cid:88) i = −∞ (cid:0) − (cid:1) i ϕ ( d i ) = k (cid:88) i = −∞ (cid:0) − (cid:1) i ϕ ( d (cid:48) i ) , where d k (cid:54) = 0 for k ≥ m (cid:54) k be the largest integer such that d m (cid:54) = d (cid:48) m , and consider the point ( − ) − m x . Recall that multiplying x by ( − ) − m is the analog of moving the decimal point m places to the left if m is positive andto the right if it is negative. Let ω := ( d k . . . d m +1 d m ) − / and ω (cid:48) := ( d (cid:48) k . . . d (cid:48) m +1 d (cid:48) m ) − / . Then ω, ω (cid:48) ∈ Z [ ] are distinct, and it follows from our assumption and the definition of the tile F that( − ) − m x − ϕ ( ω ) , ( − ) − m x − ϕ ( ω (cid:48) ) ∈ F . Hence, we obtain( − ) − m x ∈ ( F + ϕ ( ω )) ∩ ( F + ϕ ( ω (cid:48) )) . As tiles only overlap on their boundaries, this implies that x ∈ ( − ) m ∂ ( F + ϕ ( ω )). Therefore,a point x ∈ K has two different expansions if and only if x ∈ Γ , where Γ := (cid:83) m ∈ Z ( − ) m ∂ ( F + ϕ ( Z [ ])). Since Z [ ] is countable, Γ is a countable union of the sets ( − ) m ∂ ( F + ϕ ( z )), m ∈ Z , z ∈ Z [ ], each of which has measure 0. Thus µ ( Γ ) = 0, which gives the result. References [1] Shigeki Akiyama, Christiane Frougny, and Jacques Sakarovitch. Powers of rationals modulo1 and rational base number systems.
Israel J. Math. , 168:53–91, 2008.[2] Petr Ambroˇz, Daniel Dombek, Zuzana Mas´akov´a, and Edita Pelantov´a. Numbers with integerexpansion in the numeration system with negative base.
Funct. Approx. Comment. Math. ,47(part 2):241–266, 2012.[3] Val´erie Berth´e, Anne Siegel, Wolfgang Steiner, Paul Surer, and J¨org M. Thuswaldner. Fractaltiles associated with shift radix systems.
Adv. Math. , 226(1):139–175, 2011.[4] Daniel M. Gordon. A survey of fast exponentiation methods.
J. Algorithms , 27(1):129–146,1998.[5] Vittorio Gr¨unwald. Intorno all’aritmetica dei sistemi numerici a base negativa con partico-lare riguardo al sistema numerico a base negativo-decimale per lo studio delle sue analogiecoll’aritmetica (decimale).
Giornale di Matematiche di Battaglini , 23:203–221, 1885. Errata,p. 367.[6] Jan E. Holly. Pictures of ultrametric spaces, the p -adic numbers, and valued fields. Amer.Math. Monthly , 108(8):721–728, 2001.[7] John E. Hutchinson. Fractals and self-similarity.
Indiana Univ. Math. J. , 30(5):713–747,1981.[8] Aubrey J. Kempner. Anormal Systems of Numeration.
Amer. Math. Monthly , 43(10):610–617,1936.[9] Donald E. Knuth. An imaginary number system.
Comm. ACM , 3:245–247, 1960.[10] Donald E. Knuth.
The art of computer programming. Vol. 2 . Addison-Wesley, Reading, MA,1998. Seminumerical algorithms, Third edition.[11] Susan Landau. Polynomials in the nation’s service: using algebra to design the advancedencryption standard.
Amer. Math. Monthly , 111(2):89–117, 2004.[12] Alfr´ed R´enyi. Representations for real numbers and their ergodic properties.
Acta Math.Acad. Sci. Hungar. , 8:477–493, 1957. 1013] Klaus Scheicher, Paul Surer, J¨org M. Thuswaldner, and Christiaan E. van de Woestijne. Digitsystems over commutative rings.
Intern. J. Number Theory , 10(6):1459–1483, 2014.[14] Nikita Sidorov. Almost every number has a continuum of β -expansions. Amer. Math. Monthly ,110(9):838–842, 2003.[15] Anne Siegel. Repr´esentation des syst`emes dynamiques substitutifs non unimodulaires.
ErgodicTheory Dynam. Systems , 23(4):1247–1273, 2003.[16] Wolfgang Steiner and J¨org M. Thuswaldner. Rational self-affine tiles.