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Homework 8, Real Analysis

Due date: September 29

Problem 1

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

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


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

  1. $latex Y_\alpha$ is a $latex G_\delta$ subset of $latex \R$

  2. $latex X = \bigcap_{\alpha>0} Y_\alpha$ is a dense $latex G_\delta$ subset of $latex \R$.

  3. For each $latex x\in\R$, $latex x\not\in X$ iff there exists a polynomial p over $latex \R$ such that $latex p(n)d(nx)>1$ for all $latex n\ge1$.

Problem 2

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

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

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

  1. If x is an irrational algebraic number of degree $latex d>1$, then $latex x\in\mathcal D(d)$.

  2. $latex \mathscr D$ is of first category and, therefore, a typical real number is Liouville.

Problem 3

  1. If $latex A\subset X$ is connected, then so is its closure $latex \bar A$.

  2. If A is connected and $latex A \subset B \subset \bar A$, then so is B.

Problem 4

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

Problem 5

  1. If $latex f:[0,1]\to[0,1]$ is continuous, then there exists $latex c\in[0,1]$ such that $latex f(c) = c$.

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