Kaufman's dimension doubling theorem states that for a planar Brownian motion \{\mathbf{B}(t): t \in [0,1]\} we have \mathbb P(\dim \mathbf{B}(A)=2\dim A \textrm{ for all } A\subset [0,1])=1, where \dim may denote both Hausdorff dimension \dim_H and packing dimension \dim_P . The main goal of the paper is to prove similar uniform dimension results in the one-dimensional case. Let 0 < \alpha < 1 and let \{B(t): t \in [0,1]\} be a fractional Brownian motion of Hurst index \alpha . For a deterministic set D\subset [0,1] consider the following statements: (A) \mathbb P(\dim_H B(A)=(1/\alpha) \dim_H A \textrm{ for all } A\subset D)=1 , (B) \mathbb P(\dim_P B(A)=(1/\alpha) \dim_P A \textrm{ for all } A\subset D)=1 , (C) \mathbb P(\dim_P B(A)\geq (1/\alpha) \dim_H A \textrm{ for all } A\subset D)=1 . We introduce a new concept of dimension, the modified Assouad dimension, denoted by \dim_{MA} . We prove that \dim_{MA} D\leq \alpha implies (A), which enables us to reprove a restriction theorem of Angel, Balka, Máthé, and Peres. We show that if D is self-similar then (A) is equivalent to \dim_{MA} D\leq \alpha . Furthermore, if D is a set defined by digit restrictions then (A) holds if and only if \dim_{MA} D \leq \alpha or \dim_H D=0 . The characterization of (A) remains open in general. We prove that \dim_{MA} D\leq \alpha implies (B) and they are equivalent provided that D is analytic. Let D be compact, we show that (C) is equivalent to \dim_H D\leq \alpha . This implies that if \dim_H D\leq \alpha and \Gamma_D=\{E\subset B(D)\colon \dim_H E=\dim_P E\} , then \mathbb P(\dim_H (B^{-1}(E)\cap D)=\alpha \dim_H E \textrm{ for all } E\in \Gamma_D)=1. In particular, all level sets of B|_{D} have Hausdorff dimension zero almost surely.