In this paper, we study the ascending chain condition (ACC) conjecture for minimal log discrepancies (mlds), proposed by the third author. We show the ACC conjecture holds for singularities admitting $ϵ$-plt blow-ups. In particular, this gives the ACC for mlds for exceptional singularities. The key ingredients in the proofs of our main results are the Birkar–Borisov–Alexeev–Borisov theorem, proved by Birkar, the boundedness of complements conjecture for arbitrary DCC coefficients, proposed by the third author and proved in this paper, and the existence of uniform $\mathbb{R}$-complementary rational polytopes.

In this paper, we study the theory of complements, introduced by Shokurov, for Calabi–Yau type varieties with the coefficient set [0, 1]. We show that there exists a finite set of positive integers $\mathcal{N}$, such that if a threefold pair $(X/Z\ni z,B)$ has an $\mathbb{R}$-complement which is klt over a neighborhood of z, then it has an n-complement for some $n\in \mathcal{N}$. We also show the boundedness of complements for $\mathbb{R}$-complementary surface pairs.