A note on the relative growth of products of multiple partial quotients in the plane

Abstract

Let r = [a 1 (r ), a 2 (r ), . . .] be the continued fraction expansion of a real number r ∈ R. The growth properties of the products of consecutive partial quotients are tied up with the set admitting improvements to Dirichlet's theorem. Let (t 1 , . . . , t m ) ∈ R m + and let Ψ : N → (1, ∞) be a function such that Ψ(n) → ∞ as n → ∞. We calculate the Hausdorff dimension of the set of all (x, y) ∈ [0, 1) 2 such that 2020 Mathematics Subject Classification: 11K50, 11K55.