  We generalize the rationality theorem of the accumulation points of log
canonical thresholds which was proved by Hacon, M\textsuperscript{c}Kernan, and
Xu. Further, we apply the rationality to the ACC problem on the minimal log
discrepancies. We study the set of log discrepancies on varieties with fixed
Gorenstein index. As a corollary, we prove that the minimal log discrepancies
of three-dimensional canonical pairs with fixed coefficients satisfy the ACC.
