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Let ν \nu be a probability measure on S L d ( Z ) \mathrm {SL}_d(\mathbb {Z}) satisfying the moment condition E ν ( ‖ g ‖ ϵ ) > ∞ \mathbb {E}_\nu (\|g\|^\epsilon )>\infty for some ϵ \epsilon . We show that if the group generated by the support of ν \nu is large enough, in particular if this group is Zariski dense in S L d \mathrm {SL}_d , for any irrational x ∈ T d x \in \mathbb {T}^d the probability measures ν ∗ n ∗ δ x \nu ^{* n} * \delta _x tend to the uniform measure on T d \mathbb {T}^d . If in addition x x is Diophantine generic, we show this convergence is exponentially fast.

The aim of this paper is to extend recent work of S. Konyagin and the author on Gauss sum estimates for large degree to the case of ‘sparse’ polynomials. In this context we do obtain a nearly optimal result, improving on the works of Mordell and of Cochrane and Pinner. The result is optimal in terms of providing some power gain under conditions on the exponents in the polynomial that are best possible if we allow arbitrary coefficients. As in earlier work referred to above, our main combinatorial tool is a sum-product theorem. Here we need a version for product spaces F p × F p \mathbb {F}_{p}\times \mathbb {F}_{p} for which the formulation is obviously not as simple as in the F p \mathbb {F}_{p} -case. Again, the method applies more generally to provide nontrivial bounds on (possibly incomplete) exponential sums involving exponential functions. At the end of the paper, some applications of these are given to issues of uniform distribution for power generators in cryptography.

We prove the following theorem: for all positive integers $b$ there exists a positive integer $k$, such that for every finite set $A$ of integers with cardinality $|A| > 1$, we have either $$ |A + ... + A| \geq |A|^b$$ or $$ |A \cdot ... \cdot A| \geq |A|^b$$ where $A + ... + A$ and $A \cdot ... \cdot A$ are the collections of $k$-fold sums and products of elements of $A$ respectively. This is progress towards a conjecture of Erd��s and Szemer��di on sum and product sets., 33 pages, no figures, submitted, J. Amer. Math. Soc. (Proxy submission)

The main result is the following. Let Ω \Omega be a bounded Lipschitz domain in R d \mathbb {R}^{d} , d ≥ 2 d\geq 2 . Then for every f ∈ L d ( Ω ) f\in L^{d}(\Omega ) with ∫ f = 0 \int f =0 , there exists a solution u ∈ C 0 ( Ω ¯ ) ∩ W 1 , d ( Ω ) u\in C^{0}(\bar \Omega )\cap W^{1, d}(\Omega ) of the equation div u = f u=f in Ω \Omega , satisfying in addition u = 0 u=0 on ∂ Ω \partial \Omega and the estimate ‖ u ‖ L ∞ + ‖ u ‖ W 1 , d ≤ C ‖ f ‖ L d \begin{equation*}\Vert u\Vert _{L^{\infty }}+\Vert u\Vert _{W^{1, d}}\leq C\Vert f\Vert _{L^{d}} \end{equation*} where C C depends only on Ω \Omega . However one cannot choose u u depending linearly on f f . Our proof is constructive, but nonlinear—which is quite surprising for such an elementary linear PDE. When d = 2 d=2 there is a simpler proof by duality—hence nonconstructive.

Consider G(n, p) to be the probability space of random graphs on n vertices with edge probability p. We will be considering subsets of this space defined by monotone graph properties. A monotone graph property P is a property of graphs such that a) P is invariant under graph automorphisims. b) If graph H has property P , then so does any graph G having H as a subgraph. A monotone symmetric family of graphs is a family defined by such a property. One of the first observations made about random graphs by Erdos and Renyi in their seminal work on random graph theory [12] was the existence of threshold phenomena, the fact that for many interesting properties P , the probability of P appearing in G(n, p) exhibits a sharp increase at a certain critical value of the parameter p. Bollobas and Thomason proved the existence of threshold functions for all monotone set properties ([6]), and in [14] it is shown that this behavior is quite general, and that all monotone graph properties exhibit threshold behavior, i.e. the probability of their appearance increases from values very close to 0 to values close to 1 in a very small interval. More precise analysis of the size of the threshold interval is done in [7]. This threshold behavior which occurs in various settings which arise in combinatorics and computer science is an instance of the phenomenon of phase transitions which is the subject of much interest in statistical physics. One of the main questions that arises in studying phase transitions is: how “sharp” is the transition? For example, one of the motivations for this paper arose from the question of the sharpness of the phase transition for the property of satisfiability of a random kCNF Boolean formula. Nati Linial, who introduced me to this problem, suggested that although much concrete analysis was being performed on this problem the best approach would be to find general conditions for sharpness of the phase transition, answering the question posed in [14] as to the relation between the length of the threshold interval and the value of the critical probability. In this paper we indeed introduce a simple condition and prove it is sufficient. Stated roughly, in the setting of random graphs, the main theorem states that if a property has a coarse threshold, then it can be approximated by the property of having certain given graphs as a subgraph. This condition can be applied in a more

We establish global wellposedness and scattering for the H 1 H^{1} - critical defocusing NLS in 3D i u t + Δ u − u | u | 4 = 0 \begin{equation*}iu_{t}+\Delta u - u|u|^{4}=0 \end{equation*} assuming radial data ϕ ∈ H s \phi \in H^{s} , s ≥ 1 s\geq 1 . In particular, it proves global existence of classical solutions in the radial case. The same result is obtained in 4D for the equation i u t + Δ u − u | u | 2 = 0. \begin{equation*}iu_{t}+\Delta u -u|u|^{2} =0. \end{equation*}