Thomas Wolff

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The purpose of this paper is to prove an essentially sharp L2 Fourier restriction estimate for light cones, of the type which is called bilinear in the recent literature. Fix d ≥ 3, denote variables in Rd by (x, xd) with x ∈ Rd−1, and let Γ = {x : xd = |x| and 1 ≤ xd ≤ 2}. Let Γ1 and Γ2 be disjoint conical subsets, i.e. Γi = {x ∈ Γ : x xd ∈ Ωi} where Ωi are(More)
where Sn−1 is the unit sphere in R. This paper will be mainly concerned with the following issue, which is still poorly understood: what metric restrictions does the property (1) put on the set E? The original Kakeya problem was essentially whether a Kakeya set as defined above must have positive measure, and as is well-known, a counterexample was given by(More)
Let N be a large parameter, C a constant, and let ΓN (C) denote the C-neighborhood of the cone segment {ξ : 2−CN ≤ |ξ| ≤ 2N}. For fixed N , we take a partition of unity subordinate to a covering of Sd−1 by caps Θ of diameter about N− 1 2 , and use this to form a (smooth) partition of unity yΘ on ΓN(C) in the natural way. We will write ΓN,Θ(C) = supp yΘ. Let(More)
Let Λ be a compact planar set of positive finite one-dimensional Hausdorff measure. Suppose that the intersection of Λ with any rectifiable curve has zero length. Then a theorem of Besicovitch (1939) states that the orthogonal projection of Λ on almost all lines has zero length. Consequently, the probability p(Λ, ) that a needle dropped at random will fall(More)
The sharp Littlewood conjecture states that for fixed N > 1, if D(z) = 1 + z + z + •• • + z~, then on the unit circle \z\ = 1, \\D\\i is the minimum of Ц/ld for / of the form f(z)zzc0 + c1z"i+--+ cN_lz"with \ck\ = 1; more generally, \\D\\P is the min/max of ||f\\p for fixed p £ [0,2]/[2, oo]. In the paper this is proved for the special case where f(z) —(More)
The purpose of this paper is to prove some new variants on the harmonic analyst’s uncertainty principle, i.e. interplay between the support of a function and its Fourier transform, and to apply them to some questions in spectral theory. These results were suggested by work on the one dimensional Anderson model due to Campanino-Klein and others. The paper(More)
1. L 1 Fourier transform If f ∈ L 1 (R n) then its Fourier transform isˆf : R n → C defined by ˆ f(ξ) = e −2πix·ξ f (x)dx. More generally, let M(R n) be the space of finite complex-valued measures on R n with the norm µ = |µ|(R n), where |µ| is the total variation. Thus L 1 (R n) is contained in M(R n) via the identification f → µ, dµ = f dx. We can(More)
We prove several results related to a question of Steinhaus: is there a set E ⊂ R such that the image of E under each rigid motion of R contains exactly one lattice point? Assuming measurability we answer the analogous question in higher dimensions in the negative, and we improve on the known partial results in the two dimensional case. We also consider a(More)