  We provide quantitative estimates for the supremum of the Hausdorff dimension
of sets in the real line which avoid $\varepsilon$-approximations of arithmetic
progressions. Some of these estimates are in terms of Szemer\'{e}di bounds. In
particular, we answer a question of Fraser, Saito and Yu (IMRN, 2019) and
considerably improve their bounds. We also show that Hausdorff dimension is
equivalent to box or Assouad dimension for this problem, and obtain a lower
bound for Fourier dimension.
