Salem sets and restriction properties of Fourier transforms

Arithmetic Progressions in Sets of Fractional Dimension

“…In fact, the supporting measures µ = µ[k] satisfy a ball condition of the form (A), a fact that was observed in [23]. The following result is a rephrasing of Proposition 3.2 in [23], with special attention to the implicit constants.…”

“…Such estimates were first obtained by Mockenhaupt in [23], following Tomas's proof of a restriction estimate for the sphere in R n [31]. The main result in [23] is an L p (dx) → L 2 (dµ) bound for the Fourier restriction operator in the range 1 ≤ p < 2(2 − 2α + β)/(4(1 − α) + β). For our application, we are interested in values of p near the endpoint.…”

“…where 1 < q, q ′ < ∞ are exponents such that 1 q + 1 q ′ = 1, and the double integral is taken over the same region as in (23). On that region we have |η 1 − η 2 | ≤ 2R and…”

“…[28]). In the paper [23] that inspired much of our work here, Mockenhaupt proved a restriction-type result for Salem sets in R d . Specifically, he proved that if µ obeys (A)-(B), then there is a restriction estimate of the form…”

Some Recent Progress on the Restriction Conjecture

“…2 For surfaces of lower dimension, see [17], [44], [38], [1]; for fractal sets in R, see [39], [45], [38];…”