On Hausdorff dimension of radial projections

For any x ∈ Rd, d ≥ 2, denote by πx : Rd\{x} → Sd-1 the radial projection Given a Borel set E ⊂ Rd with dimH E ≤ d-1, in this paper we investigate for how many x ∈ Rd the radial projection πx preserves the Hausdorff dimension of E, namely whether dimH πx(E) = dimH E. We develop a general framework to link πx(E), x ∈ F, and πy(F), y ∈ E, for any Borel set F ⊂ Rd. In particular, this allows us to apply Orponen fs estimate on visibility to study whether dimH πx(E) = dimH E for some x ∈ F. More precisely, we show for any Borel set E ⊂ Rd with dimH E ∈ (d-2, d-1]. This improves the Peres.Schlag bound when dimH E ∈ (d-3/2, d-1], and it is optimal at the endpoint dimH E = d-1.