A famous conjecture of Littlewood (c. 1930) concerns approximating two real numbers by rationals the same denominator, multiplying errors. In a lesser-known paper, Wang and Yu (Chin Ann Math 2:1–12, 1981) established an asymptotic formula for number such approximations, valid almost always. Using quantitative Koukoulopoulos–Maynard theorem Aistleitner–Borda–Hauke, together with bounds arising f...

Let $$Q \rightarrow R$$ be a surjective homomorphism of Noetherian rings such that Q is Gorenstein and R as Q-bimodule admits finite resolution by modules which are projective on both sides. We define an adjoint pair functors between the homotopy category totally acyclic R-complexes Q-complexes. This analogous to classical module categories Q. As consequence, we obtain precise notion approximat...