Let $��\in(1,2)$ and $x\in [0,\frac{1}{��-1}]$. We call a sequence $(��_{i})_{i=1}^\infty\in\{0,1\}^{\mathbb{N}}$ a $��$-expansion for $x$ if $x=\sum_{i=1}^{\infty}��_{i}��^{-i}$. We call a finite sequence $(��_{i})_{i=1}^{n}\in\{0,1\}^{n}$ an $n$-prefix for $x$ if it can be extended to form a $��$-expansion of $x$. In this paper we study how good an approximation is provided by the set of $n$-prefixes. Given $��:\mathbb{N}\to\mathbb{R}_{\geq 0}$, we introduce the following subset of $\mathbb{R}$, $$W_��(��):=\bigcap_{m=1}^{\infty}\bigcup_{n=m}^\infty\bigcup_{(��_{i})_{i=1}^{n}\in\{0,1\}^{n}}\Big[\sum_{i=1}^{n}\frac{��_i}{��^{i}}, \sum_{i=1}^n\frac{��_i} {��^i}+��(n)\Big]$$ In other words, $W_��(��)$ is the set of $x\in\mathbb{R}$ for which there exists infinitely many solutions to the inequalities $$0\leq x-\sum_{i=1}^{n}\frac{��_{i}}{��^{i}}\leq ��(n).$$ When $\sum_{n=1}^{\infty}2^{n}��(n)