On Hausdorff dimension of projections

Untitled

“…Theorem 8.5 extends to smooth functions a result of Mattila [35] (generalizing earlier results of Marstrand [34] and Kaufman [31]) that makes the same conclusion for almost every linear function from R n to R m , in the sense of Lebesgue measure on the space of m-by-n matrices. Theorems 8.5 and 8.6 and their predecessors follow from a potential-theoretic characterization of the dimensions involved.…”

On restricted families of projections in ℝ3

“…This does not mean that results sharper than the ones by Marstrand and Mattila are not possible -and indeed they are: numerous such improvements have appeared since 1954. A particularly elegant one is due to R. Kaufman [7] from 1968: given 0 < s < 1, a family L of lines through the origin in R 2 such that dim H {L ∩ S 1 : L ∈ L} = s, and a set B ⊂ R 2 with dim H B < s, one may always find a line L ∈ L such that the projection of B into L has dimension dim H B. Moreover, the result is sharp in the sense that it may fail if dim H B = s. This was shown by Kaufman and Mattila [8] in 1975: for any 0 < s < 1, they managed to construct an s-dimensional set B ⊂ R 2 , the dimension of the projections of which drops strictly below s for a certain s-dimensional family L = L B of lines through the origin.…”

“…Unless otherwise stated, (ρ θ ) θ∈U and (π θ ) θ∈U will always stand for non-degenerate families of projections onto lines and planes, respectively. When dim H B lies on certain intervals, dimension conservation for non-degenerate families of projections can be proven directly using the classical 'potential theoretic' method pioneered by R. Kaufman in [7]. These bounds are the content of the following proposition.…”

The effect of projections on dimension in the Heisenberg group

“…We shall employ potential theoretic methods first used in this context by Kaufman in [12] and later generalized in [16]. There have been many studies on Marstrand type projection results.…”