Averages of Gauss map iterations in arithmetic progressions

Abstract The theory of uniform Diophantine approximation concerns the study of Dirichlet improvable (or non-improvable) numbers. The metrical aspect of this theory leads to the study of the products of partial quotients in continued fractions. It is known that the dimension of the set of Dirichlet non-improvable numbers depends upon the number of partial quotients in the product string. However, the Hausdorff dimension is the same when partial quotients in the product are separated with a gap of fixed length. In this paper, we provide a detailed Hausdorff dimension analysis of the set of real numbers with a large product of partial quotients with indices in arithmetic progressions. This poses significant challenges as opposed to the considerations of consecutive partial quotients. As an application of our general theorem, in particular, we obtain the Hausdorff dimension of the set $$\begin{aligned} E(\psi ):=\left\{ x\in [0, 1): \root n \of {a_{n}(x)a_{2n}(x)\cdots a_{n^2}(x)}\ge \psi (n) \ \mathrm{for \ infinitely \ many} \ n\in \mathbb {N}\right\} . \end{aligned}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mtable> <mml:mtr> <mml:mtd> <mml:mrow> <mml:mi>E</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>ψ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>:</mml:mo> <mml:mo>=</mml:mo> <mml:mfenced> <mml:mi>x</mml:mi> <mml:mo>∈</mml:mo> <mml:mrow> <mml:mo>[</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>:</mml:mo> <mml:mroot> <mml:mrow> <mml:msub> <mml:mi>a</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:msub> <mml:mi>a</mml:mi> <mml:mrow> <mml:mn>2</mml:mn> <mml:mi>n</mml:mi> </mml:mrow> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>⋯</mml:mo> <mml:msub> <mml:mi>a</mml:mi> <mml:msup> <mml:mi>n</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> <mml:mi>n</mml:mi> </mml:mroot> <mml:mo>≥</mml:mo> <mml:mi>ψ</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mspace/> <mml:mrow> <mml:mi>for</mml:mi> <mml:mspace/> <mml:mi>infinitely</mml:mi> <mml:mspace/> <mml:mi>many</mml:mi> </mml:mrow> <mml:mspace/> <mml:mi>n</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>N</mml:mi> </mml:mfenced> <mml:mo>.</mml:mo> </mml:mrow> </mml:mtd> </mml:mtr> </mml:mtable> </mml:mrow> </mml:math>