Metrical properties of exponentially growing partial quotients

Abstract A fundamental challenge within the metric theory of continued fractions involves quantifying sets of real numbers especially when their partial quotients exhibit specific growth rates. For any positive function Φ, the Wang–Wu theorem (2008) comprehensively describes the Hausdorff dimension of the set <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi mathvariant="script">E</m:mi> <m:mn>1</m:mn> </m:msub> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi mathvariant="normal">Φ</m:mi> <m:mo rspace="0.278em" stretchy="false">)</m:mo> </m:mrow> <m:mo rspace="0em">:</m:mo> <m:mo lspace="0em">=</m:mo> <m:mrow> <m:mo stretchy="false">{</m:mo> <m:mi>x</m:mi> <m:mo>∈</m:mo> <m:mrow> <m:mo stretchy="false">[</m:mo> <m:mn>0</m:mn> <m:mo>,</m:mo> <m:mn>1</m:mn> <m:mo rspace="0.278em" stretchy="false">)</m:mo> </m:mrow> <m:mo rspace="0.278em">:</m:mo> <m:msub> <m:mi>a</m:mi> <m:mi>n</m:mi> </m:msub> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>x</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mo>≥</m:mo> <m:mi mathvariant="normal">Φ</m:mi> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>n</m:mi> <m:mo rspace="0.500em" stretchy="false">)</m:mo> </m:mrow> <m:mtext>for infinitely many</m:mtext> <m:mi>n</m:mi> <m:mo>∈</m:mo> <m:mi mathvariant="double-struck">N</m:mi> <m:mo stretchy="false">}</m:mo> </m:mrow> <m:mo lspace="0em">.</m:mo> </m:mrow> </m:math> \mathcal{E}_{1}(\Phi):=\{x\in[0,1):a_{n}(x)\geq\Phi(n)\ \text{for infinitely many}\ n\in\mathbb{N}\}. Various generalisations of this set exist, such as substituting one partial quotient with the product of consecutive partial quotients in the aforementioned set which has connections with the improvements to Dirichlet’s theorem, and many other sets of similar nature. Establishing the upper bound of the Hausdorff dimension of such sets is significantly easier than proving the lower bound. In this paper, we present a unified approach to get an optimal lower bound for many known setups, including results by Wang–Wu [ Adv. Math. (2008)], Huang–Wu–Xu [ Israel J. Math. (2020)], Tan–Zhou [ Nonlinearity (2023)], and several others. We also provide a new theorem derived as an application of our main result. We do this by finding an exact Hausdorff dimension of the set <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:msub> <m:mi>S</m:mi> <m:mi>m</m:mi> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:msub> <m:mi>A</m:mi> <m:mn>0</m:mn> </m:msub> <m:mo>,</m:mo> <m:mi mathvariant="normal">…</m:mi> <m:mo>,</m:mo> <m:msub> <m:mi>A</m:mi> <m:mrow> <m:mi>m</m:mi> <m:mo>−</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msub> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mo>⁢</m:mo> <m:mover accent="true"> <m:mo>=</m:mo> <m:mo>def</m:mo> </m:mover> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">{</m:mo> <m:mrow> <m:mi>x</m:mi> <m:mo>∈</m:mo> <m:mrow> <m:mo stretchy="false">[</m:mo> <m:mn>0</m:mn> <m:mo>,</m:mo> <m:mn>1</m:mn> <m:mo rspace="0.278em" stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo rspace="0.278em">:</m:mo> <m:mrow> <m:mrow> <m:mrow> <m:msub> <m:mi>c</m:mi> <m:mi>i</m:mi> </m:msub> <m:mo>⁢</m:mo> <m:msubsup> <m:mi>A</m:mi> <m:mi>i</m:mi> <m:mi>n</m:mi> </m:msubsup> </m:mrow> <m:mo>≤</m:mo> <m:mrow> <m:msub> <m:mi>a</m:mi> <m:mrow> <m:mi>n</m:mi> <m:mo>+</m:mo> <m:mi>i</m:mi> </m:mrow> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>x</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo>&lt;</m:mo> <m:mrow> <m:mn>2</m:mn> <m:mo>⁢</m:mo> <m:msub> <m:mi>c</m:mi> <m:mi>i</m:mi> </m:msub> <m:mo>⁢</m:mo> <m:msubsup> <m:mi>A</m:mi> <m:mi>i</m:mi> <m:mi>n</m:mi> </m:msubsup> </m:mrow> </m:mrow> <m:mo>,</m:mo> <m:mr