Hausdorff dimension and distance sets

According to a result of K. Falconer (1985), the setD(A)={|x−y|;x, y ∈A} of distances for a Souslin setA of ℝ ⁿ has positive 1-dimensional measure provided the Hausdorff dimension ofA is larger than (n+1)/2.* We give an improvement of this statement in dimensionsn=2,n=3. The method is based on the fine theory of Fourier restriction phenomena to spheres. Variants of it permit further improvements which we don’t plan to describe here. This research originated from some discussions with P. Mattila on the subject. dimA >n/2 would be the optimal result forn ≥ 2.     

Pointwise ergodic theorems for arithmetic sets

With an appendix on return-time sequences     

A nonlinear version of Roth's theorem for sets of positive density in the real line

It is proved that given ε>0, there is δ(ε)>0 such that ifS is a measurable set of [0,N], |S|>εN, then there is a triplex, x+h, x+h ² inS withh satisfyingh>δ(ε)N ^1/2. The argument is related to [B] and uses the behavior of certain non-linear convolution-type operators. The method can be adapted in a variety of situations. For instance, it can be used to prove the analogue of the previous statement with the square replaced by another power,h ³,h ⁴ etc.     

Construction of sets of positive measure not containing an affine image of a given infinite structure

This paper deals with the problem of existence of infinite structures in euclidean space such that every set of positive measure contains an affine image of it. We contribute to P. Erdös’ question about sequences in the real line, by showing that no triple sum of infinite sets has this property.     

Estimation of certain exponential sums arising in complexity theory

It is shown that the correlation on ${\{0,1\}}^n$ between parity and a polynomial $p(x_1,\dots ,x_n)\in \mathbb{Z}[X_1,\dots ,X_n](\mathrm{mod}q)$, q a fixed odd number and $p(X)$ of degree d arbitrary but fixed, is exponentially small in n as $n\to \infty$. An application to circuit complexity, from where the problem originates, is given. To cite this article: J. Bourgain, C. R. Acad. Sci. Paris, Ser. I 340 (2005).