A new version of distributional chaos and the relations between distributional chaos in a sequence and other concepts of chaos

The paper’s Theorem 1 states that on a compact dynamical system, if Q has bounded gaps, then DC1 is equivalent to SDC along Q, and proves it via elementary counting and uniform continuity of finitely many iterates; the candidate solution uses the same ingredients with a cleaner block-partition argument. The paper’s necessity direction proceeds by comparing counts up to n versus up to q_n and shows the Q-liminf/limsup match the natural ones, while the sufficiency direction uses uniform continuity for j ∈ {1,…,M} to propagate closeness across blocks and again compares counts; these are exactly the steps the candidate formalizes with sharper inequalities and explicit use of q_n ≥ n and q_n ≤ q_0 + M n. See the theorem statement and proof outline in the PDF (Theorem 1) and its definitions (DC1 and SDC) and key proof devices (finite-iterate uniform continuity and block counting) . A remark in the paper also clarifies why unbounded gaps can break the equivalence, aligning with the model’s emphasis on the bounded-gap hypothesis .