Number Through a Group Theoretic Lens Hyman Bass University of Michigan Klein Project Session Mathfest Pittsburgh, August 7,

Euclidean Division,or Division with Remainder Given a, b in R, a ≠ 0, there exist unique real q, r with b=qa + r q in Z and0 ≤ r ≤ |a| -a 0 a 2a... (q-1)a qa b (q+1)a _.______.______._____.__...____.______._____.___.______.____ | | q = q a (b) = [b/a]andr = r a (b) 2

Additive groups of (real) numbers A set A of real numbers containing 0 and closed under + & -. Examples: - R Q Z - Z m={qm | q in Z }e.g. {0} “Cyclic group” - Per(f)={ p in R | f(x+p) = f(x) for all x} E.g. Per(sin)= Z 2π 3

Discrete Additive Groups A is discrete if, for some ε> 0, |a| ≥ εfor all a ≠ 0 in A. In this case, |a – b| ≥ εfor all a ≠ b in A. Examples: Cyclic groups are discrete Q and R are not 4

The Discrete/Dense Dichotomy 5 T HEOREM An additive group of real numbers is either discrete or dense in R Suppose that A is not discrete. To show A is dense means that given x in R and ε > 0, we can find a in A with |a – x| < ε. By assumption there is a c ≠ 0 in A with |c| < ε. By Euclidean Division, we can write x = qc + r with q in Z and 0 ≤ r < |c| < ε. Then a = qc is in A, and |x – a| = r < ε. COROLLARY If f(x) is continuous and non-constant, then Per(f) is discrete.

Inclusion and Divisibility Equivalent conditions: b belongs to Z a Z b ≤ Z a b = qa for some integer q Terminology and notation: a divides b b is a(n integer) multiple of a a | b 6

Intersection and Sum A and B are additive groups: is an additive group, the largest one contained in both A and B A + B = { a + b | a in A and b in B} is an additive group, the smallest containing both A and B (The union is not an additive group unless one contains the other) 7

THEOREM Discrete additive groups are cyclic In fact, if A is discrete, there is a unique a ≥ 0 such that A = Z a. If A = {0} take a = 0. Otherwise A contains elements > 0, and then discreteness implies that there is a least one, call it a. (Proof) Then if b is in A, Euclidean Division gives b = qa + r with 0 ≤ r 0 in A implies that r = 0, so b = qa is in Z a. 8 Question: When is Za + Zb discrete?

Commensurability Equivalent conditions on non-zero real numbers a and b: b belongs to Q a Q b = Q a a/b is rational a and b are both integer multiples of some d > 0 9

Commensurability Theorem Equivalent conditions on non-zero real numbers a and b: 1. Z a + Z b is discrete 2. Z a intersect Z b is not zero 3.a and b are commensurable In this case, d = gcd(a, b) ≥ 0 and m = lcm(a b) ≥ 0 are defined by Z d= Z a + Z b Z m= Z a intersect Z b PROPOSITION ab = dm 10

√2 is irrational, so... {a + b√2 | a, b in Z } is dense in R Diagonally cut the unit square into two triangles. Starting at 0, lay copies of one end-to-end along the line with a leg on the line, and lay copies of the other along the line with the hypotenuse on the line. Consider the sets of meeting points of each sequence of triangles. Then they have no points in common, but come arbitrarily close to each other. 11

If gcd(a, b) = 1 and a | bc, then a | c Primes and unique factorization theorem gcd(ca, cb)=|c|gcd(a, b) gcd(a, b + qa)=gcd(a, b) Euclidean Algorithm to find gcd(a, b) 12

Multiplicative groups of (non-zero real) numbers Examples: R x Q x { ± 1}(0, ∞) a Z = {a q | q an integer}(“Cyclic”) What is a discrete multiplicative group? M is discrete if there is an open interval around 1 whose intersection with M is {1}. 13

New Phenomena Torsion: -1 Direct products: R x ={±1} x (0, ∞) Homomorphisms: Sgn : R x  {± 1} |. |: R x  (0, ∞) Question : What are the discrete multiplicative groups in R x ?In (0, ∞)? 14

THEOREM Discrete multiplicative subgroups of (0, ∞) are cyclic Proof: Continuous isomorphisms exp: R  (0, ∞) log: (0, ∞)  R M ≤ (0, ∞) discrete. A = log(M) is discrete in R, hence A = Z a. Then M = exp(A) = exp( Z a) = m Z 15

General discrete multiplicative groups M a multiplicative group. M >0 = M intersect (0, ∞) 1.M is discrete iff M >0 is discrete iff |M| is discrete 2.If -1 is in M then M >0 = |M| and M = {±1} x |M| 3.If -1 is not in M and M ≠ M >0, then M = a Z for some a < 0. 16

Discrete groups of Rotations Let Rot denote the group (wrt composition) of rotations of the plane. It consists of rotations ρ(a), rotation about the origin by an angle of 2πa radians. Thus we have a homomorphism ρ: R  Rotρ(a+b) = ρ(a) o ρ (b) and ρ( Z ) = I (= ρ(0)). Let M be a discrete subgroup of Rot. Then it is easy to see that A = ρ -1 (M) is a discrete subgroup of R, containing Z. Then A = Z (1/n) for a unique n in N, and so M is cyclic generated by ρ(1/n). 17

Further Developments: Additive and multiplicative groups of... C (Connections with plane linear algebra and geometry) Z/Z m(Modular arithmetic, Chinese Remainder Theorem, etc.) 18

Thank you 19