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.
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.