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Friday, May 4, 2012

Additive subgroups of the reals and irrational rotation

We present a useful characterization of the additive subgroups of \mathbb{R}.

Theorem. Let G be a subgroup of (\mathbb{R}, +). Then precisely one of the following holds:
(i) G = d\mathbb{Z} for some d \in \mathbb{R};
(ii) G is dense in \mathbb{R}.

Proof: If G = \{0\}, then G = 0\mathbb{Z}. Assume G is nontrivial. Let d = \inf\{g: 0 < g \in G\}. Note that d \geq 0 since for every nonzero y \in G, ky \in G and ky > 0 for some k \in \mathbb{Z}.
Case 1: d > 0. We claim that d \in G. If not, then for every \epsilon > 0 there exists g \in G with g \in (d, d + \epsilon). It follows that for every \epsilon > 0 there are g, h \in G such that 0 < g - h < \epsilon, thus there exists x \in G with 0 < x < \epsilon (let x = g - h). This contradicts to d > 0. We also have d = \min\{|g|: 0 \neq g \in G\}. Now fix any 0< g \in G and consider S = \{g - nd: n = 0, 1, 2, \ldots\}. The set S is discrete, we can find n \geq 0 such that g - nd \geq 0 and g - (n + 1)d < 0. Since g - nd, g - (n +1)d \in G, g - (n + 1)d \leq -d, so g - nd \leq 0 and g = nd. Therefore G = d\mathbb{Z}.
Case 2: d = 0. Arguing as in the beginning of case 1, we find that 0 is a cluster point of G \cap \mathbb{R}_+. Let (a, b) be any nonempty bounded open interval of \mathbb{R}. Take \epsilon = b - a and pick some g \in G \cap (0, \epsilon). For sufficiently large n, ng \geq a, and we take the smallest such n. Then ng - g < a, i.e. ng \in (a - g, a], thus ng + g \in (a, a + g] \subseteq (a, b). Therefore G \cap (a, b) \neq \emptyset. It follows that G is dense in \mathbb{R}.

The above result can be used to prove the density of the orbit of an irrational rotation in the unit circle.

Corollary. (i) For an irrational t, \mathbb{Z} + t\mathbb{Z} is dense in \mathbb{R}.
(ii) For an irrational t, \{e^{2\pi int}: n \in \mathbb{Z}\} is dense in the unit circle.

Proof: (i) Note that \mathbb{Z} + t\mathbb{Z} is a subgroup of (\mathbb{R}, +). If \mathbb{Z} + t\mathbb{Z} = a\mathbb{Z} for some a \in \mathbb{R}, then 1 = an for some n \in \mathbb{Z}, so a \in \mathbb{Q}. But t = am for some m \in \mathbb{Z}, so t \in \mathbb{Q}, contradiction. By Theorem, \mathbb{Z} + t\mathbb{Z} must be dense in \mathbb{R}.
(ii)  Let e^{2\pi i x} be a point on the unit circle. By (i), there exist sequences m_k, n_k of integers such that |m_k + n_k t - x| \rightarrow 0. Then |e^{2 \pi i n_k t} - e^{2 \pi i x}| = |e^{2\pi i x}||e^{2 \pi i (n_k t - x)} - 1| = |e^{2\pi i (m_k + n_k t - x)} - 1| = O(|m_k + n_k t - x|) \rightarrow 0. Hence \{e^{2\pi int}\}_{n \in \mathbb{Z}} is dense in the unit circle.

Wednesday, May 2, 2012

Hadamard gap theorem

Definition. A sequence \{n_k\} of nonnegative integers is said to be lacunary if there is some \delta > 0 such that n_{k + 1}/n_k \geq 1 + \delta for all k \geq 1.

A typical example of a lacunary sequence is n_k = 2^k.

Theorem (Hadamard Gap Theorem).  Suppose \{n_k\} is a lacunary sequence and the power series f(z) = \sum_{n = 0}^\infty a_n z^n with a_n nonzero precisely when n = n_k for some k and has radius of convergence 1. Then f(z) cannot be analytically extended to any larger region containing a point on the unit circle.

Proof: Suppose f(z) extends to an analytic function on a region containing the open unit disk B and a point w_0 on the unit circle. WLOG, we may assume w_0 = 1. Then f(z) is analytic on a region \Omega containing B \cup \{1\}. Define g(w) = f(w^m(1 + w)/2) where m is a positive integer. Since 1^m(1 + 1)/2 = 1 and |w^m(1+w)/2| = |w|^m|1+w|/2 < |w|^m \leq 1 when |w| \leq 1 and w \neq 1 because |1 + w| < 2, so g(w) is well-defined when |w| \leq 1, hence when |w| < 1 + \epsilon for some \epsilon > 0. Write g(w) = \sum_{m = 0}^\infty b_m w^m for |w| < 1 + \epsilon. Note that the powers of w in the terms of [w^m(1 + w)/2]^{n_k} range from n_k m to n_k (m + 1). Since \{n_k\} is lacunary, we can choose m to be so large that n_{k + 1} m > n_k (m + 1) for all k, then the powers of w in all of [w^m(1 + w)/2]^{n_k} are distinct. Now
\sum_{n = 0}^{n_k} a_n z^n = \sum_{m = 0}^{n_k (m + 1)} b_m w^m.
When |w| < 1 + \epsilon, RHS converges as k \rightarrow \infty. But LHS is the partial sums of \sum_{k = 1}^\infty a_{n_k} z^{n_k} as k varies, so the power series converges for all z = w^m(1 + w)/2 where |w| < 1 + \epsilon, which includes the case z = 1. The image of B(0, 1 + \epsilon) under the map w^m(1 + w)/2 is an region W containing 1, so the power series of f(z) at 0 converges on W, contradicting its radius of convergence being 1. QED.

Reference:
Gamelin - Complex Analysis
Rudin - Real and Complex Analysis (3rd Ed.)

Tuesday, May 1, 2012

Some results in measure theory

Proposition. Let \mu be a positive Borel measure on \mathbb{R} and \epsilon > 0. Then for almost every x, we have \int_\mathbb{R} \frac{d\mu(t)}{|x - t|^{1 + \epsilon}} = +\infty.
Proof: For each positive integer N, define E_N = \{x \in \mathbb{R}: \int_\mathbb{R} d\mu(t) / |x - t|^{1 + \epsilon} \leq N\}. Let I be any bounded closed interval with E_N \cap I nonempty. Pick x \in E_N \cap I and write I = [x - \delta_1, x + \delta_2]. We have N \geq \int_\mathbb{R} \frac{d\mu(t)}{|x - t|^{1 + \epsilon}} \geq \int_I \frac{d\mu(t)}{|x - t|^{1 + \epsilon}} \geq \frac{\mu(I)}{\max\{\delta_1, \delta_2\}^{1 + \epsilon}} \geq \frac{\mu(E_N \cap I)}{(\delta_1 + \delta_2)^{1 + \epsilon}}, i.e. \mu(E_N \cap I) \leq Nm(I)^{1 + \epsilon} where m(I) is the length of the interval I. Now let [a, b] \subseteq \mathbb{R} and n \in \mathbb{N}. Decompose [a, b] into a union of n closed subintervals of length (b - a)/n, then \mu(E_N \cap [a, b]) \leq nN\left(\frac{b - a}{n}\right)^{1 + \epsilon} \rightarrow 0 as n \rightarrow \infty. Hence \mu(E_N \cap [a, b]) = 0. It follows that \mu(E_N) = 0 for all N \in \mathbb{N} and our result follows.

(To be continued)