This paper discusses the uniform distribution of sequences modulo a subdivision. A sequence $ (x_n) $ is uniformly distributed modulo a subdivision $ z $ if the normalized sequence $ \{x_n\}_z $ is uniformly distributed modulo 1. The authors prove a generalization of known results on the uniform distribution of sequences $ (n\theta) $ modulo a subdivision $ z $, using an elementary method. The paper defines "almost uniformly distributed" sequences modulo $ z $, which are sequences for which there exists an infinite sequence of integers $ N_1 < N_2 < \cdots $ such that the normalized sequence $ \{x_n\}_z $ is uniformly distributed for these $ N_i $. The main theorem states that if $ \theta $ is a positive real number and $ z = (z_n) $ is an increasing sequence of real numbers with $ z_0 = 0 $ and $ z_n/n \to \infty $ as $ n \to \infty $, then the sequence $ (x_n) = (\theta n) $ is almost uniformly distributed modulo $ z $. The proof involves analyzing the distribution of $ x_n $ within intervals defined by the subdivision $ z $, and showing that the normalized sequence $ \{x_n\}_z $ converges to the uniform distribution modulo 1. The result is shown to hold under the condition that $ z_n/z_{n-1} \to 1 $ as $ n \to \infty $, which implies that the subdivision becomes finer as $ n $ increases. The paper also references several related results and theorems in the literature on uniform distribution modulo a subdivision.This paper discusses the uniform distribution of sequences modulo a subdivision. A sequence $ (x_n) $ is uniformly distributed modulo a subdivision $ z $ if the normalized sequence $ \{x_n\}_z $ is uniformly distributed modulo 1. The authors prove a generalization of known results on the uniform distribution of sequences $ (n\theta) $ modulo a subdivision $ z $, using an elementary method. The paper defines "almost uniformly distributed" sequences modulo $ z $, which are sequences for which there exists an infinite sequence of integers $ N_1 < N_2 < \cdots $ such that the normalized sequence $ \{x_n\}_z $ is uniformly distributed for these $ N_i $. The main theorem states that if $ \theta $ is a positive real number and $ z = (z_n) $ is an increasing sequence of real numbers with $ z_0 = 0 $ and $ z_n/n \to \infty $ as $ n \to \infty $, then the sequence $ (x_n) = (\theta n) $ is almost uniformly distributed modulo $ z $. The proof involves analyzing the distribution of $ x_n $ within intervals defined by the subdivision $ z $, and showing that the normalized sequence $ \{x_n\}_z $ converges to the uniform distribution modulo 1. The result is shown to hold under the condition that $ z_n/z_{n-1} \to 1 $ as $ n \to \infty $, which implies that the subdivision becomes finer as $ n $ increases. The paper also references several related results and theorems in the literature on uniform distribution modulo a subdivision.