Euclidean Models of the p-adic Integers

Euclidean Models of the p-adic Integers Scott Zinzer December 12, 2012 1 Introduction Much of our visual perception is based in what seems to be standard Euclidean space; we can easily imagine perfectly straight lines and planes, and we can even create realistic drawings of the world we perceive using straight lines. Whether the physical universe is actually Euclidean in its geometry is open for debate, but it is hard to deny our reliance on Euclidean geometry for perceiving the space in which we immediately exist. Certainly, this is enforced by the prevalence of Euclidean geometry in the standard elementary mathematics education curriculum. So strong is our reliance on Euclidean geometry that even in the setting of general (potentially non-euclidean) spaces, we often attempt to create Euclidean pictures to help us better understand the spaces under consideration. Even the words we use in these general settings seem inspired by our experience with Euclidean spaces (e.g. balls, disks, and spheres in metric spaces). Kurt Hensel first discovered 1 the p-adic numbers in the late 19th century (here p is a prime number). The p-adic world provides a canvas on which algebra, analysis, and number theory interplay to create a landscape quite unlike anything expected from our immediate experience with Euclidean space. Topologically, the ring Z p of p-adic integers provides a mess of counterexamples to standard misconceptions in metric space topology (phrases such as open ball and closed ball become meaningless), although the average non number-theorist may never have a formal introduction to the p-adic integers or to the p-adic topology. There are several ways of motivating and constructing the p-adic numbers algebraically or analytically. In this paper, we will simply construct the p-adic integers from the rational integers (Z) by introducing the p-adic 1 or invented, depending on one s philosophical view of mathematics. 1

metric on Z and completing. Although most of the algebraic construction will be supressed, there is one striking fact worth mentioning. The ring of p-adic integers is a subring of the field of p-adic numbers, the field obtained by completing the rational numbers with respect to the p-adic absolute value and its induced metric. Ostrowski s theorem states that any (non-trivial) absolute value on Q is equivalent to the usual (real) absolute value or to some p-adic absolute value. In this way, the only completions of Q with respect to metrics inherited from non-trivial absolute values on Q are the field of real numbers (R) or some p-adic field. In this way, the p-adic numbers actually arise just as naturally from Q as does R. For many number-theoretical applications, the p-adic metrics are much more useful than the usual (real) absolute value, and we actually come to adopt several conventions in order to treat R on the same footing as the p-adic completions of Q. The primary goal of this paper is to attempt to describe methods of visualizing the p-adic integers within familiar Euclidean space. To do so, we seek subsets of n-dimensional Euclidean space that are homeomorphic to the p-adic integers that we can describe by an inductive construction. These subsets turn out to be fractal subsets of Euclidean space, and this perhaps helps to highlight just how novel the p-adic integers are in a topological sense. 1.1 Notation Z, Q, R, and C will denote the ring of rational integers, the field of rational numbers, the field of real numbers, and the field of complex numbers, as usual. p will always denote a prime number in Z + and will often be fixed in what follows. 2 Background 2.1 Ultrametric Spaces The standard introductory references for the following material include [2, 4]. The material in this first subsection is motivated especially by [6]. See also [5, 3]. Recall that metric on a set X is a function d : X X R satisfying 1. d(x, y) 0 for all x, y X. 2. d(x, y) = 0 if and only if x = y. 2

3. d(x, y) = d(y, x) for all x, y X. 4. d(x, y) d(x, z) + d(z, y) for all x, y, z X. If X is a set with metric d, then the pair (X, d) is called a metric space. Property 4 above of a metric d is often endearingly referred to as the triangle inequality, in analogy with the corresponding property of the lengths of the sides of Euclidean triangles. We strengthen property 4 to the following: 4. d(x, y) max{d(x, z), d(z, y)} for all x, y, z X. The new property 4 is called the strong triangle inequality (sometimes also the ultrametric inequality) 2. Clearly, any function d satisfying 4 also satisfies 4, and this leads us to our first definition: Definition 1. Let (X, d) be a metric space. The metric d is called an ultrametric 3 and (X, d) is called an ultrametric space if d satisfies the strong triangle inequality. Obviously, all of the properties of metric spaces carry over for ultrametric spaces, but the strong triangle inequality introduces several notable topological properties not holding in general for metric spaces. As usual, given an ultrametric space X, a point x X, and r > 0, the open ball of radius r centered at a is B r (x) = {y X : d(x, y) < r} and the closed ball of radius r centered at x is B r (x) = {y X : d(x, y) r} Our notation is not meant to imply that B r (a) is the topological closure of B r (a), and the terminology is not meant to imply openness or closedness with respect to the topology on X. When we refer simply to a ball in X, we mean a set of one of the above forms. Lemma 1. Let (X, d) be an ultrametric space. 2 For this reason, I will often refer to the usual triangle inequality as the weak triangle inequality. 3 In the setting of p-adic analysis, the ultrametric is sometimes also called non- Archimedean, although this term is most often reserved for the absolute value that arises from the p-adic valuation. Of course, this absolute value induces the p-adic metric. 3

1. Let x, y, z X with d(x, z) d(y, z). Then d(x, y) = max{d(x, z), d(y, z)} (all triangles are isosceles). 2. Any ball in X is both open and closed. 3. Any point of a ball may be its center. 4. If B 1 and B 2 are balls in X which are not disjoint, then B 1 B 2 or B 2 B 1. Proof. 1. Without loss of generality, suppose d(x, z) < d(y, z). Then But also, d(x, y) max{d(x, z), d(y, z)} = d(y, z) d(x, z) < d(y, z) max{d(x, y), d(x, z)} Thus, we must have d(y, z) d(x, y). 2. Let x X and r > 0. Of course, B r (x) is open and B r (x) is closed. Suppose y X \ B r (x). Then d(x, y) r. If d(y, z) < r, then d(y, z) < d(x, y). Thus d(x, z) = max{d(x, y), d(y, z)} = d(x, y) r This gives B r (y) X\B r (x), so B r (x) is closed. Finally, let y B r (x). If z B r (y), then d(z, x) max{d(z, y), d(x, y)} r so B r (y) B r (x) and B r (x) is open. 3. First consider a ball of the form B r (x) and let y B r (x). If z B r (y), then d(z, x) max{d(z, y), d(y, x)} < r so that B r (y) B r (x). A symmetrical argument gives the reverse containment, so B r (x) = B r (y). An identical argument works for a ball of the form B r (x). 4. If neither B 1 B 2 nor B 2 B 1, choose x B 1 \ B 2, y B 2 \ B 1 and z B 1 B 2. From the previous part, z is a center of both B 1 and B 2. Therefore d(z, x) > d(z, y) since x B 1 and y B 1 and d(z, x) < d(z, y) since y B 2 and x B 2. This is impossible. 4

We will need to discuss completions of ultrametric spaces in the next section, so we record the following: Lemma 2. Let (X, d) be an ultrametic space and (x n ) a sequence in X. Then (x n ) is Cauchy if and only if for all ε > 0 there is N > 0 so that d(x n, x n+1 ) < ε for all n N. Proof. Obviously any Cauchy sequence satisfies this property. Now let (x n ) be any sequence in X satisfying this property and let ε > 0. Choose N > 0 so that d(x n, x n+1 ) < ε for all n N. Now let m > n N. Then d(x m, x n ) max{d(x n, x n+1 ), d(x n+1, x n+2 ),, d(x m 1, x m )} < ε. 2.2 The p-adic Integers Let p Z be a fixed prime. Any element x Z \ {0} can be written in the form x = p e y, where y is relatively prime to p. By unique factorization in the integers, the exponent of p which appears in this factorization is unique. Definition 2. For x Z \ {0}, the p-adic valuation of x is the non-negative integer v p (x) such that x = p vp(x) y with y relatively prime to p. It is relatively straightforward to see that v p satisfies the following properties: 1. v p (x) 0 for all x Z \ {0}. 2. v p (xy) = v p (x) + v p (y) for all x, y Z \ {0}. 3. v p (x + y) min{v p (x), v p (y)} for all x, y Z \ {0}. We extend v p to all of Z by putting v p (0) = ; then v p : Z Z 0 { } still satisfies 1-3 above on all of Z. We use the p-adic valuation to define a metric on Z. Definition 3. The p-adic topology on Z is the metric topology with the p- adic metric d : Z Z R given by d(x, y) = p vp(x y) (where p = 0) 4. With the p-adic topology, Z is an ultrametric space. 4 In essence, the p-adic metric is measuring distance in terms of congruence modulo the prime p. Integers that are congruent modulo a high power of p have a difference with a large p-adic valuation, and hence are assigned a small distance by the p-adic metric. 5

Hereafter, we always endow Z with the p-adic topology. Notice that {d(x, y) : x, y Z} = {0} {p n : n 0} Definition 4. For a metric space (X, d), the metric d is called discrete if x 1, x 2,... X, y 1, y 2,... X with d(x 1, y 1 ) > d(x 2, y 2 ) > implies d(x n, y n ) 0. The p-adic metric on Z is discrete. All of the general properties of ultrametric spaces above hold for Z with the p-adic topology, but unfortunately, Z ( is not complete with respect to this topology. For instance, the sequence n k=0 pk) is p-adic Cauchy but does not converge p-adically in Z. n 0 Definition 5. The p-adic integers Z p is the completion of Z with respect to the p-adic topology. Then Z p is a complete ultrametric space and Z is dense in Z p, making Z p a separable ultrametric space. Note that we still have {d(x, y) : x, y Z p } = {0} {p n : n 0} so that the p-adic metric on Z p is discrete. In fact, N is dense in Z p and there is a particularly useful way of representing any element of Z p as a limit of a sequence in N. Recall that each element a N has a unique finite base-p expansion a = a 0 + a 1 p + a 2 p 2 + + a N p N with a i {0,..., p 1}. Lemma 3. Let α Z p. Then α can be written uniquely as α = i 0 a i p i with a i {0,..., p 1} 5. We call this the p-adic expansion of α. From this result, it follows that Z p is uncountable. We use the p-adic expansion to prove the following. Lemma 4. Z p is sequentially compact, hence compact. 5 There are other possible choices for the p-adic digits a i, but {0,..., p 1} will suffice for our purposes. 6

Proof. Let (α n ) be a sequence in Z p. For each n, we write α n = i 0 a (n) i p i By the pigeonhole principle, there is b 0 {0,..., p 1} for which a (n) 0 = b 0 for infinitely many n. The collection of all such terms of (α n ) yields a subsequence (α 0n ), the terms of which all have b 0 as the first digit of their p-adic expansions. We now repeat this construction inductively to obtain a sequence of subsequences of (α n ), ((α kn ) n ) k with (α kn ) n a subsequence of (α k+1,n ) n and a p-adic integer b = k 0 b k p k such that for each k, every term of (α kn ) n agrees with b in its first k + 1 digits. Then the diagonal sequence (α kk ) is a subsequence of (α n ) which converges to b. It is easy to determine the distance between two p-adic integers if their p-adic digits are known. If the first n digits of x, y Z p are identical, then the first n digits of x y are all 0, which means that p n divides x y, so that d(x, y) p n. Consider now any closed ball in Z p, say B ε (x). Then there is n 0 so that B ε (x) = B 1/p n(x). Now let y Z have the same initial n p-adic digits as x (y is given by truncating the p-adic expansion of x). Then B 1/p n(y) = B ε (x) and we can apply a similar argument to an open ball in Z p. Therefore, Lemma 5. The collection of balls in Z p is countable. Thus, Z p is a second countable metric space. In light of the above, we will always write a ball in Z p as a ball with a radius of the form p n for some n centered at a Z with 0 a p n 1. We now change notation 6 and let a + p n Z p = B 1/p n(a) The properties of balls in ultrametric spaces implies p 1 a + p n Z p = (a + bp n ) + p n+1 Z p b=0 6 This change reflects the algebraic structure of Z p. 7

and in particular, Finally, we record p 1 Z p = b + pz p b=0 Lemma 6. Z p is totally disconnected. Proof. Let a Z p and let A Z p be any connected subset containing a. Suppose there is b A with b a. Choose 0 < ε < d(a, b). Then B ε (a) is both open and closed in Z p, hence B ε (a) A, A is both open and closed in A. This is impossible, so it must be that A = {a}. This result implies that Z p is not locally Euclidean, so is certainly not a real manifold (although it is Hausdorff and second countable) 7. For this paper, we are primarily concerned with Z p as a topological space (a metric space), so we neglect many of the algebraic properties of Z p, although some of these properties have already come into play behind the scenes. For the sake of completeness we cite some useful algebraic facts. Fact 1. The following are true: 1. Z p is a (topological) ring and the addition and multiplication of p-adic integers are the only continuous operations on Z p extending addition and multiplication on Z. The ring operations can be defined on p-adic expansions and are described by polynomials in finitely many variables. 2. Z p is a principal ideal domain and its nontrivial ideals are precisely the ideals generated by p k. 3. Z p is a local ring with unique maximal ideal pz p. Therefore, Z p = Z p \ pz p. 4. Z p /pz p = Z/pZ. Taking for granted that Z p is a ring, we observe that for any x, y Z p d(px, py) = p vp(px py) = p vp(p(x y)) = p vp(p) vp(x y) = p 1 p vp(x y) = p 1 d(x, y) 7 Although the title of this course is The Geometry and Topology of Manifolds, I still hoped this topic would be appreciated and would be of general interest. 8

so that multiplication by p is a contraction mapping. important in the following section. This fact will be 3 Euclidean Models of Z p The material in this section is inspired by the related material in [5] and [3]. The paper [1] is also a source of motivation. The ultimate goal is to embed Z p topologically into Euclidean space. The fact that this can be done is perhaps a bit surprising. We remark however, that although we can topologically embed Z p into Euclidean space, we cannot do so while preserving algebraic structure. Definition 6. A subset E R n which is homeomorphic to Z p is called a Euclidean model of Z p. A subset E R which is homeomorphic to Z p is called a linear model of Z p. In what follows, we will describe some Euclidean models of Z p and iterative constructions to visualize them. To find such subsets of R n, the general approach will be to exploit the decomposition of Z p as a disjoint union of open balls of the form a + pz p with a {0,..., p 1} and to use the fact that multiplication by p is a contraction mapping (so that pz p looks like a shrunken copy of Z p ). 3.1 Cantor Sets We begin with a somewhat unsatisfactory Euclidean model of Z p ; namely, we show that there is essentially only one family of linear models of Z p. However, there is one interesting fact that results from this consideration. Recall that the (classical) Cantor set is a subset of the unit inverval I = [0, 1] R obtained from the following iterative construction. From C 0 = I = [0, 1], delete the open middle third (1/3, 2/3) to obtain C 1 = [0, 1/3] [2/3, 1], a union of two closed intervals. To obtain C n+1 from C n, where C n is a union of closed intervals, delete the open middle third of each closed interval. Then each C n consists of 2 n closed intervals of length 3 n (so is compact) and C n+1 C n. The Cantor set C is C = C n [0, 1] with the subset (metric) topology inherited from R. Lemma 7. The Cantor set C has the following properties. 9

1. C is compact. 2. C is uncountable. 3. C is totally disconnected. 4. C is perfect, meaning it is closed and does not contain isolated points. Properties 1-3 should sound familiar; they are topological properties of Z p that we highlighted in the previous section. In fact, there is a (classification) theorem about subsets of R satisfying the above properties. Theorem 1. Any compact, perfect, totally disconnected subset of R is homeomorphic to the Cantor set. This theorem implies that any linear model of Z p must be homeomorphic to the Cantor set. We now show that the classical Cantor set C is a linear model of Z 2, the 2-adic integers, and then describe Cantor-like subsets which are linear models of Z p. To describe points of C, we adopt a base-3 numeration system with digits {0, 1, 2} for points in [0, 1]. Then the set C n in the iterative construction of C is the subset of [0, 1] whose first n digits in this base-3 expansion are not 1. Therefore, C consists of all numbers in [0, 1] with base-3 expansions c i 3 i i 1 with c i {0, 1}, and any element of C can be uniquely expressed in this form. This looks promising, since there are only two choices for 2-adic digits for 2-adic expansions of elements of Z 2. We define Φ : Z 2 C via Φ : i 0 a i 2 i (2a i )3 (i+1) i 0 This certainly gives a bijection Z 2 C by the uniqueness of the expansions, but more is true. Lemma 8. Φ : Z 2 C is a homeomorphism. Proof. Let x, y Z 2 with p-adic expansions x = i 0 x i 2 i, y = i 0 y i 2 i Then d(x, y) 2 n if and only if x i = y i for all i n. But this occurs if and only if 2x i = 2y i for all i n, which occurs if and only of Φ(x) and Φ(y) are in the same component of C n, which is equivalent to Φ(x) Φ(y) 3 n. This gives the continuity of both Φ and Φ 1. 10

For linear models of Z p, we generalize the construction of the Cantor set slightly. For a prime number p, the generalized p-cantor set C (p) is a subset of the unit inverval I = [0, 1] R obtained from the following iterative construction. Divide C (p) 0 = I = [0, 1] into 2p 1 equal subintervals and delete every second open interval to obtain C (p) 1 To obtain C (p), where C (p), a union of closed intervals. n+1 from C(p) n n is a union of closed intervals, divide each closed interval in 2p 1 equal subintervals and delete every second open interval. Then C (p) = C n (p) is uncountable, compact, and perfect. If we adopt a base-(2p 1) numeration system with digits {0, 1,..., 2p 2} for [0, 1], then C n is the subset of [0, 1] whose first n digits are even. Then C (p) consists of all numbers in [0, 1] with expansions c i (2p 1) i i 1 with c i {0, 2, 4,... 2p 2}, and any element of C (p) can be uniquely expressed in this form. Lemma 9. For any p, the map Φ (p) : Z p C (p) given by Φ (p) : i 0 a i p i (2a i )(2p 1) (i+1) i 0 is a homeomorphism. The proof of this result is analogous to the proof for Z 2 and C: two p-adic integers are close if and only if their images under Φ (p) lie in the same connected component of some approximating set C n (p), which occurs if and only if the images are close with respect to the Euclidean metric on R. Since C and C (p) are homeomorphic, we obtain the following mildly surprising result. Lemma 10. Z p and Z q are homeomorphic for all p, q. In particular, the subsets of R n we obtain in the following section are all homeomorphic. 3.2 Non-linear Euclidean Models of Z p We now describe subsets of R n (n 2) which are homeomorphic to Z p. For fixed p, let ν : {0,..., p 1} R n be an injective map, and put 11

S = ν({0,..., p 1}) R n. Define ψ b : Z p R n by ψ b : i 0 a i p i i 0 (b 1) b i+1 ν(a i) Then ψ b (Z p ) is contained in the convex hull of S. Recall that Z p = p 1 a=0 a + pz p Then ψ b (Z p ) = p 1 a=0 = v S ( (b 1) b ( (b 1) b ν(a) + 1 ) b ψ b(z p ) v + 1 ) b ψ b(z p ) i.e. ψ b (Z p ) is a union of translates of dilated copies of ψ b (Z p ). When b is chosen large enough, in fact ψ b is injective and ψ b (Z p ) is also a disjoint union of self-similar images, i.e. a fractal subset of R n. Let F = ψ b (Z p ), with b chosen large enough so that F is a disjoint union of self-similar images. We can describe an iterative process for constructing F. Let K 0 be the convex hull of S in R n. Then F K 0, so F = ( (b 1) v + 1 ) b b F v S ( (b 1) v + 1 ) b b K 0 := K 1 v S We now induct to get compact subsets K n with F = K n We can consider each K n as an approximation of F. In this sense, two p-adic integers are within 1/p n of each other if and only if their images under ψ b are in the same component of K n. As before, this gives the continuity of both ψ b and ψ 1 b when ψ b is injective. 12

3.2.1 Examples in R 2 Finally, for the sake of illustration, we describe two nice examples in the Euclidean plane using regular polygons. For these examples, we obtain highly symmetrical fractal subsets of R 2 which are Euclidean models of Z p. 8 1. Take p > 3. Consider the regular (p 1)-gon P in R 2 with vertices v 1,..., v p 1, lying on the boundary of the unit disk in R 2. We define ν : {0,..., p 1} R 2 by ν(0) = 0 and ν(i) = v i for i 0. For example, with p = 5, we take v 1 = (1, 0) and b = 4. Then ψ 4 : Z 5 R 2 is injective and F = ψ 4 (Z 5 ) R 2 is a Euclidean model of Z 5 contained within a square centered at the origin in R 2. Figure 1: The approximating superset K 4 of the fractal subset F = ψ 4 (Z 5 ). With p = 7, we again take v 1 = (1, 0) and b = 4. Then ψ 4 : Z 7 R 2 is injective and F = ψ 4 (Z 7 ) R 2 is a Euclidean model of Z 7 contained within a regular hexagon centered at the origin in R 2. 8 All images were created with care and unwavering determination by the thoughtful and skilled Marc Hester. 13

Figure 2: The approximating superset K 4 of the fractal subset F = ψ 4 (Z 7 ). 2. For p 3 consider the regular p-gon P in R 2 with vertices v 0,..., v p 1 lying on the boundary of the unit disk in R 2. We define ν : {0,..., p 1} R 2 by ν(i) = v i and take b = p. For example, with p = 3, we take v 0 = (0, 1). Then ψ 3 : Z 3 R 2 is injective and F = ψ 3 (Z 3 ) R 2 is a Euclidean model of Z 3 contained within a regular triangle centered at the origin in R 2. Figure 3: The approximating superset K 5 of the fractal subset F = ψ 3 (Z 3 ). 14

4 Conclusion For the non-number theorist (and perhaps even for some number theorists), the topological space Z p may remain an eccentric example within metric space topology, while for others (including the author) Z p is perhaps the most natural example of a metric space. We have seen how to embed Z p into Euclidean space and thus have described how we may visualize Z p within the familiar context and setting of R n. This perhaps provides additional motivation for introducing and studying the p-adic world. 5 Addendum: Euclidean models of Q p In the preceding section, we described some Euclidean models of Z p. In these cases, we obtained bounded subsets of Euclidean space, reflecting the fact that the p-adic metric is bounded. For this paper, it will suffice to define Q p as the field of fractions of Z p, i.e. Q p is obtained from Z p by inverting all non-zero elements of Z p and is the smallest field (up to isomorphism) containing Z p as a subring 9 Can we describe a similar construction for Q p to obtain fractal subsets of R n which are homeomorphic to Q p? 10 Recall that the group of invertible elements of Z p is Z p = Z p \pz p, so we need only invert p in order to obtain Q p. Elements of Q p have expansions of the form α = a i p i i k with only finitely many terms having negative exponent. In fact, we may write Q p = n 0 p n Z p We simply extend the maps ψ b to Q p in the obvious way: ψ b : a i p i i k i k (b 1) b i+1 ν(a i) Since multiplication by p is a contraction mapping, multiplication by 1 p is a dilation. From the representation of Q p as a union, we can see that ψ b (Q p ) is 9 Alternatively, we could have constructed Q p first as the completion of Q with respect to the p-adic absolute value on Q defined in terms of the p-adic valuation, which we may extend from Z to all of Q. 10 This section was added following my presentation. 15

an unbounded fractal subset of R n whose self-similarity structure is identical to that of ψ b (Z p ). We can iteratively build approximations of ψ b (Q p ) from ψ b (Z p ) by arranging dilated copies of ψ b (Z p ) appropriately within R n. References [1] Albert A. Cuoco. Visualizing the p-adic integers. Amer. Math. Monthly, 98(4):355 364, 1991. [2] Fernando Q. Gouvêa. p-adic Numbers. An Introduction. Universitext, 1997. [3] Svetlana Katok. p-adic Analysis Compared with Real. American Mathematical Society, 2007. [4] Neal Koblitz. p-adic Numbers, p-adic Analysis and Zeta-Functions. Springer-Verlag, 1984. [5] Alain M. Robert. A Course in p-adic Analysis. Springer-Verlag, 2000. [6] W. H. Schikhof. Ultrametric Calculus. An Introduction to p-adic Analysis. Cambridge University Press, 2006. 16