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\]
Week 9 Coursework Test
- The matrix $A = [\begin{smallmatrix} 2 & 1 \\ 1 & 1 \end{smallmatrix}]$ has eigenvalues $0 \l \eta \l 1 \l \theta$. Let $v = [\begin{smallmatrix} v_1 \\ v_2 \end{smallmatrix}]$ be an eigenvector of $A$ with eigenvalue $\eta$.
- Prove for every $r,s \in \Z$ with $(r,s) \ne (0,0)$ and every $(x,y) \in [0,1)^2$ that
\[
\lim_{T \to \infty} \dfrac{1}{T} \int\limits_0^T e^{2 \pi i r(x + tv_1) + 2 \pi i s (y+tv_2)} \intd t = 0
\]
by calculating the above Riemann integral and then evaluating the limit.
- By analogy with our work on irrational rotations, postulate a uniform distribution statement that would follow from the above result.
- Consider a short line segment $L$ in $[0,1)^2$ parallel to $v$. Fix a rectangle $R \subset [0,1)^2$. With
\[
T(x,y) = (2x+y \bmod 1, x+y \bmod 1)
\]
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
\[
\left\{ \left[ \dfrac{a}{10^r}, \dfrac{a+1}{10^r} \right) : r \in \N, 0 \le a \l 10^r \right\}
\]
of subsets of $[0,1)$ generates the Borel σ-algebra on $[0,1)$.