A fine property of sets at points of Lebesgue density.

Abstract. Consider an integer n ≥ 2, m ∈ [n,+∞) and put α := m−1 n−1 . Moreover: • Let γx : (0,+∞)→Rn be the line through x = (x1, . . . , xn) ∈ Rn defined as γx (t) := (tx1, tαx2, . . . , tαxn); • If T is any motion of Rn, then let γ (T ) x be the line through x = (x1, . . . , xn) ∈ Rn defined as γ (T ) x := T ◦ γT −1(x); • Let H1 denote the one-dimensional Hausdorff measure in Rn. The main goal of this paper is to prove the following property: If x0 is an m-density point of a Lebesgue measurable set E and T is an arbitrary motion of Rn mapping the origin to x0, then we have lim sup t→0+ H1(E ∩ γ (T ) x ((0, t])) H1(γ (T ) x ((0, t])) = 1 for almost every x ∈ T ({1} × Rn−1). An