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We establish the dimension version of Falconer's distance set conjecture forsets of equal Hausdorff and packing dimension (in particular, forAhlfors-regular sets) in all ambient dimensions. In dimensions $d=2$ or $3$, weobtain the first explicit estimates for the dimensions of distance sets ofgeneral Borel sets of dimension $d/2$; for example, we show that the set ofdistances spanned by a planar Borel set of Hausdorff dimension $1$ hasHausdorff dimension at least $(\sqrt{5}-1)/2\approx 0.618$. In higherdimensions we obtain explicit estimates for the lower Minkowski dimension ofthe distance sets of sets of dimension $d/2$. These results rely on newestimates for the dimensions of radial projections that may have independentinterest.

On the distance sets spanned by sets of dimension $d/2$ in $\mathbb{R}^d$