Week 9 Coursework Test

The matrix $A = [\begin{matrix} 2 & 1 \\ 1 & 1 \end{matrix}]$ has eigenvalues $0 < η < 1 < θ$ . Let $v = [\begin{matrix} v_{1} \\ v_{2} \end{matrix}]$ be an eigenvector of $A$ with eigenvalue $η$ .
1. Prove for every $r, s \in Z$ with $(r, s) \neq (0, 0)$ and every $(x, y) \in [0, 1)^{2}$ that $lim_{T \to \infty} \frac{1}{T} \int_{0}^{T} e^{2 π i r (x + t v_{1}) + 2 π i s (y + t v_{2})} d t = 0$ by calculating the above Riemann integral and then evaluating the limit.
2. By analogy with our work on irrational rotations, postulate a uniform distribution statement that would follow from the above result.
3. Consider a short line segment $L$ in $[0, 1)^{2}$ parallel to $v$ . Fix a rectangle $R \subset [0, 1)^{2}$ . With $T (x, y) = (2 x + y mod 1, x + y mod 1)$ describe what $T^{- n} (L)$ looks like when $n$ is large. Using your postulate, what proportion of $T^{- n} (L)$ do you expect belongs to $R$ as $n$ increases?
Prove the collection ${[\frac{a}{10^{r}}, \frac{a + 1}{10^{r}}) : r \in N, 0 \leq a < 10^{r}}$ of subsets of $[0, 1)$ generates the Borel σ-algebra on $[0, 1)$ .