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## 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 $f(\xi) = f(-\xi)$.