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Let A and B be Borel subsets of the Euclidean n-space with dim A + dim B > n and let 0 < u < dim A + dim B − n where dim denotes Hausdorff dimension. Let E be the set of those orthogonal transformations g ∈ O(n) for which dim A ∩ (g(B) + z) ≤ u for almost all z ∈ R n . If dim A + dim B > n + 1, then dim E ≤ n(n − 1)/2 + 1 − u, and if dim A ≤ (n − 1)/2, then dim E ≤ n(n−1)/2−u. If A is a Salem set and 0 < u < dim A+dim B−n and dim A+dim B > 2n−1, then dim A ∩ (B + z) > u for z in a set ofdoi:10.5186/aasfm.2017.4236 fatcat:4mg4h3kycncdtgtbc6f2o7yr34