Homework 8, Real Analysis

Due date: September 29

Problem 1

Let d(x) = d(x,\Z) denote the distance from x\in\R to the nearest integer. For q\in\Z_+, \alpha>0, define the sets

U_\alpha(q) = \{x\in\R: d(qx)< q^{-\alpha}\}

and

Y_\alpha = \{x\in\R: x belongs to infinitely many U_\alpha(q)\}.

  1. Y_\alpha is a G_\delta subset of \R
  2. X = \bigcap_{\alpha>0} Y_\alpha is a dense G_\delta subset of \R.
  3. For each x\in\R, x\not\in X iff there exists a polynomial p over \R such that p(n)d(nx)>1 for all n\ge1.

Problem 2

We say that a real number x is Diophantine of exponent \alpha >0 if there exists a constant c>0 such that

\Big| x - \dfrac{p}{q} \Big| > \dfrac{c}{q^\alpha}

for all rationals p/q. We denote by \mathcal D(\alpha) the set of Diophantine numbers of exponent \alpha and \mathscr D = \bigcup_\alpha \mathcal D(\alpha) the collection of all Diophantine numbers. A Liouville number is a number which is neither rational nor Diophantine. Let \mathscr L the set of Liouville numbers.

  1. If x is an irrational algebraic number of degree d>1, then x\in\mathcal D(d).
  2. \mathscr D is of first category and, therefore, a typical real number is Liouville.

Problem 3

  1. If A\subset X is connected, then so is its closure \bar A.
  2. If A is connected and A \subset B \subset \bar A, then so is B.

Problem 4

X is connected iff every continuous f:X\to Y into a discrete space is constant.

Problem 5

  1. If f:[0,1]\to[0,1] is continuous, then there exists c\in[0,1] such that f(c) = c.
  2. If f:\mathbb S^1\to\R is continuous, then there exists \xi\in\mathbb S^1 such that f(\xi) = f(-\xi).
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