In this paper we present a deterministic Cantor-type construction of linear fractal Salem sets with prescribed dimension. The construction rests on a paper of Kaufman [10] where he investigated the Fourier dimension of the set of α-well approximable numbers in$R$.

The main facts about Hausdorff and packing measures and dimensions of a Borel set$E$are revisited, using$determining$set functions $\phi_\alpha\colon\mathcal{B}_E\to(0,\infty)$ , where $\mathcal{B}_E$ is the family of all balls centred on$E$and α is a real parameter. With mild assumptions on φ_{α}, we verify that the main density results hold, as well as the basic properties of the corresponding box dimension. Given a bounded open set$V$in ℝ^{$D$}, these notions are used to introduce the$interior$and$exterior$measures and dimensions of any Borel subset of ∂$V$. We stress that these dimensions depend on the choice of φ_{α}. Two determining functions are considered, φ_{α}($B$)=Vol_{$D$}($B$∩$V$)diam($B$)^{α-$D$}and φ_{α}($B$)=Vol_{$D$}($B$∩$V$)^{α/$D$}, where Vol_{$D$}denotes the$D$-dimensional volume.

We give a holomorphic extension result for continuous CR functions on a non-generic CR submanifold$N$of ℂ^{$n$}to complex transversal wedges with edges containing$N$. We show that given any$v$∈ℂ^{$n$}∖($T$_{$p$}$N$+$iT$_{$p$}$N$), there exists a wedge of direction$v$whose edge contains a neighborhood of$p$in$N$, such that any continuous CR function defined locally near$p$extends holomorphically to that wedge.

Roughly speaking, to any space M with perfect obstruc- tion theory we associate a space N with symmetric perfect obstruction theory. It is a cone over M given by the dual of the obstruction sheaf of M, and contains M as its zero section. It is locally the critical locus of a function. More precisely, in the language of derived algebraic geometry, to any quasi-smooth space M we associate its (−1)-shifted cotangent bundle N. By localising from N to its C∗-fixed locus M this gives five notions of virtual signed Euler characteristic of M: (1) TheCiocan-Fontanine-Kapranov/Fantechi-G ̈ottschesignedvirtual Euler characteristic of M defined using its own obstruction theory, (2) Graber-Pandharipande’s virtual Atiyah-Bott localisation of the virtual cycle of N to M, (3) Behrend’s Kai-weighted Euler characteristic localisation of the vir- tual cycle of N to M, (4) Kiem-Li’s cosection localisation of the virtual cycle of N to M, (5) (−1)vd times by the topological Euler characteristic of M. Our main result is that (1)=(2) and (3)=(4)=(5). The first two are deformation invariant while the last three are not.

In this paper, we show that there is a Cantor set of initial conditions in the planar 4‑body problem such that all four bodies escape to infinity in a finite time, avoiding collisions. This proves the Painlevé conjecture for the 4‑body case, and thus settles the last open case of the conjecture.