This paper studies the convergence exponent of the partial quotients in the backward continued fraction (BCF) expansion of real numbers. Let \(\tau (x)\) be the convergence exponent of the BCF partial quotients of a real number \(x\in [0,1)\) . For any \(0\le \xi \le \infty \) , we investigate the size and structure of the level set \(\begin{aligned} \Delta (\xi ):= \left\{ x\in [0,1): \tau (x)=\xi \right\} \end{aligned}\) from the perspectives of Lebesgue measure, Baire category, and multifractal analysis. We prove that the set \(\Delta (\infty )\) has full Lebesgue measure and is residual, whereas for any finite \(\xi \) , the set \(\Delta (\xi )\) has Lebesgue measure zero and is of first category. Furthermore, we determine the multifractal spectrum \(\xi \mapsto \dim _\textrm{H}\Delta (\xi )\) and show that it equals 1/2 for all finite \(\xi \) and jumps to 1 at \(\xi =\infty \) . Restricting to the set \(\Lambda \) of points with non-decreasing BCF partial quotients, we derive explicit formulas for the Hausdorff dimension of generalized growth-level sets, thereby obtaining a continuous and non-increasing spectrum for \(\dim _{\textrm{H}}(\Delta (\xi ) \cap \Lambda )\) .
