Math 8601 University of Minnesota Math 8601 University of Minnesota September 9, 2009 School of Mathematics September 9, 2009 School of Mathematics

Real Analysis Real Analysis

Wikipedia Folland

Real analysis is a branch of mathematical analysis dealing with the set of real numbers. The name “real analysis” is something of an anachronism. Originally Analysis has its beginnings in the rigorous formulation of calculus. applied to the theory of functions of a real variable, it has come to encompass several subjects of a more general and abstract nature Calculus (Latin, calculus, a small stone used for counting) is a branch that underlie much of modern analysis. … [These include] measure of mathematics that includes the study of limits, derivatives, integrals, and integration theory, point set topology, and functional analysis … . and infinite series, and constitutes a major part of modern university education.

Math 8601/02 = measure and integration theory, point set Math 8601/02 = Freshman Calculus redone abstractly and topology, functional analysis … rigorously.

Math 8601 University of Minnesota Math 8601 University of Minnesota September 9, 2009 School of Mathematics September 9, 2009 School of Mathematics

Numbers Numbers

Why do we need integers? To solve the equation …==the set of positive integers 1,2,3, {} x +=10. = the set of integers Why do we need rational numbers? = the set of rati onal numb ers To solve the equation = the set of real numbers 210.x −= = the set of complex numbers Why do we need algebraic numbers? To solve the equation x2 −=20.

Math 8601 University of Minnesota Math 8601 University of Minnesota September 9, 2009 School of Mathematics September 9, 2009 School of Mathematics

Numbers Fibonacci Sequence

Why do we need complex numbers? 1,1, 2,3,5,8,13, 21,34,… To solve the equation aa01= ==+1, annn+− 1 aa 1 x2 +=10. aa nn+11=+1 − Why do we need real numbers? aann To take limits a Let xn==n , 1,2,K . n n ⎛⎞1 an−1 lim⎜⎟ 1+ n→∞ ⎝⎠n 1 Then xn+1 =+ 1 and to sum series xn 1 ∞ limx ==+ϕϕ , where 1 , 1 n . n→∞ ϕ ∑ n! n=0 orϕϕ2 −−= 1 0 , the "golden ratio".

1 Math 8601 University of Minnesota Math 8601 University of Minnesota September 9, 2009 School of Mathematics September 9, 2009 School of Mathematics Golden Ratio Golden Ratio

11 ϕ ϕϕϕϕ=1,1,10.+=+−−=2 1 1+ ϕ 1 1+ 1+

1 Approximants 113 ϕϕ≈=1, ϕϕ ≈=+= 1 2, ϕϕ ≈=+ 1 = , 12 31 121+ 1 15 8 13 21 ϕϕ≈=+1,,,, = ϕϕ ≈= ϕϕ ≈= ϕϕ ≈= … ϕ −1 1 45671 1+ 35 8 13 1 ϕ 1 1+ =−−=,10ϕϕ2 1 11ϕ − Note Fibonacci numbers.

Math 8601 University of Minnesota Math 8601 University of Minnesota September 9, 2009 School of Mathematics September 9, 2009 School of Mathematics Golden Ratio Farey Sequences

12312523++ Farey Sequence of order n : Arrange all the rational numbers with ϕϕ==,, ϕ ==== , ϕ , 1211211312 3++ 4 denominators less than or equal to n (in lowest terms) in order. 835+++ 1358 21813 ϕϕ==,, = = ϕ = = ,… 56 7 011 01121 0111231 523++ 8 35 1358 + nn==2: , ,; 3: , , , ,; n = 4: , , , , , ,; 121 13231 1432341 acac+ If ϕϕϕ===,,, and if , then . kkk−11+ 01112132341 bdbd+ n = 5:, ,,, ,, ,,, ,; 15435253451 Exercise ac 0 1111213234 51 If < (in lowest terms), and if there is no rational number between n = 6:, ,,,,,,,,,, ,; bd 1654352534561 ac ac+ and with denominator no greater than max()bd , , then is the 0111121231432534561 bd bd+ n = 7:,,,,,,,,,,,,,,,,,,; ac 1 7654 735 7 2 753 7 456 71 rational number between and with the smallest denominator. bd 0 1111121 323143 5253456 71 n = 8:, ,,,,,,, ,,,,,, ,,,,,,, ,. c.f. Farey Sequence 1 8765473 857275 8374567 81

Math 8601 University of Minnesota Math 8601 University of Minnesota September 9, 2009 School of Mathematics September 9, 2009 School of Mathematics Farey Sequences Farey Sequences

Exercise 0.5 ac If and are adjacent numbers in a Farey sequence, then bd (1)ad−=± bc 1, and ac (2) the next number to appear between and in a higher bd ac+ order Farey sequence is . bd+

0 0 1 1 1 1 1 2 1 3 2 3 1 4 3 5 2 5 3 4 5 6 7 1 8 7 6 5 4 7 3 8 5 7 2 7 5 8 3 7 4 5 6 7 8

The Farey sequence of order 8

2 Math 8601 University of Minnesota Math 8601 University of Minnesota September 9, 2009 School of Mathematics September 9, 2009 School of Mathematics Farey Sequences Farey Sequences

Exercise

ac If and are adjacent numbers in a Farey sequence, and if bd 1 a Cx is a circle of diameter tangent to the -axis at , and if 1 bb2 1 c 1 Cx is a circle of diameter tangent to the -axis at , then 2 2 dd2 d

(1)C1 and C2 are tangent, and 1 ac+ 1 (2) the circle of diameter tangent to the x -axis at b2 ()bd+ 2 bd+ a ac+ c is tangent to both CC12 and . b bd+ d

Math 8601 University of Minnesota Math 8601 University of Minnesota September 9, 2009 School of Mathematics September 9, 2009 School of Mathematics Farey Sequences Diophantine Approximation

p 0.5 For each rational ∈>[] 0,1 and for C 0 define the interval q ⎛⎞pCpC ICpq()=−⎜⎟22, − , and let ⎝⎠qqqq

VICCpq= U ( ). pq∈[]0,1

Note that VC is an open subset of [] 0,1 containing all the rationals. 1 If CV≥ , then contains all numbers in [] 0,1 . 2 C What about smaller values of C ? 0 0 1 1 1 1 1 2 1 3 2 3 1 4 3 5 2 5 3 4 5 6 7 1 8 7 6 5 4 7 3 8 5 7 2 7 5 8 3 7 4 5 6 7 8 Does VC contain the golden ratio ϕ ?

The Farey sequence of order 8

Math 8601 University of Minnesota Math 8601 University of Minnesota September 9, 2009 School of Mathematics September 9, 2009 School of Mathematics Diophantine Approximation Diophantine Approximation

Let fx( ) =−− x2 x 1. Then f(ϕ ) = 0, and Exercise 2 If yny is an algebraic number of degree , i.e., is the zero of an ⎛⎞pp ⎛⎞ ⎛⎞ pp2 ⎛ pp ⎞⎛ ⎞ fff⎜⎟=−=−−−−−=−+− ⎜⎟()ϕϕϕϕϕ ⎜⎟11() ⎜ ⎟⎜ 1. ⎟ irreducible polynomial of degree nC , then there is a > 0 such that ⎝⎠qq ⎝⎠ ⎝⎠ qq ⎝ qq ⎠⎝ ⎠ pC p ⎛⎞pp p y −≥n , for every rational . fM⎜⎟≤∈≤ −ϕ , for ∈[1 , 2 ]. qq q ⎝⎠qq q

222⎛⎞p But qf⎜⎟=−−≥ p pqq 1, so ⎝⎠q pp1 Mq2 −≥ϕϕ1, so −≥ , qqMq2

ϕ ∉V1 M

3 Math 8601 University of Minnesota Math 8601 University of Minnesota September 9, 2009 School of Mathematics September 9, 2009 School of Mathematics Diophantine Approximation Diophantine Approximation

A Little Measure Theory

p As before, for each rational ∈>[] 0,1 and for C 0 define the interval 2C μ ()ICpq()= n . q q ⎛⎞pCpC For each qq> 1, there are at most rationals in the interval [] 0,1 ICpq( ) =−⎜⎟nn,, − and let ⎝⎠qqqq with denominator equal to q . ∞∞ VICCpq= U (). 21C pq∈[]0,1 Therefore μ ()VqCC ≤=∑∑nn2−1 < ∞≥ for n3. qq==11qq Note that V is an open subset of [] 0,1 containing all the rationals. C Thus we have constructed an open set of arbitrarily small For small CV , misses a lot of algebraic numbers. C measure containing all the rationals in an interval. What else does it miss?

Math 8601 University of Minnesota Math 8601 University of Minnesota September 9, 2009 School of Mathematics September 9, 2009 School of Mathematics Liouville Numbers Morals

Definition A Liouville number is a real number xn such that, for every positive integer , p there exists a rational number , with q > 1 , such that q The real numbers have a lot of structure. p 1 0.<−x

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