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We consider the Cauchy problem for the 1-dimensional periodic cubic nonlinear Schrödinger (NLS) equation with initial data below . In particular, we exhibit nonlinear smoothing when the initial data are randomized. Then, we prove local well-posedness of the NLS equation almost surely for the initial data in the support of the canonical Gaussian measures on for each , and global well-posedness for each .
In this article, we prove the following conjecture by Lubotzky. Let , where is a local field of characteristic and where is a simply connected, absolutely almost simple -group of -rank at least 2. We give the rate of growth of
where if and only if there is an abstract automorphism of such that . We also study the rate of subgroup growth of any lattice in . As a result, we show that these two functions have the same rate of growth, which proves Lubotzky’s conjecture. Along the way, we also study the rate of growth of the number of equivalence classes of maximal lattices in with covolume at most .
We describe all the discrete subgroups of that act transitively on the set of vertices of , the Bruhat–Tits building of a pair of a characteristic nonarchimedean local field, and a simply connected, absolutely almost simple -group if is of dimension at least . In fact, we classify all such maximal subgroups. We show that there are exactly eleven families of such subgroups and explicitly construct them. Moreover, we show that four of these families act simply transitively on the vertices. In particular, we show that there is no such action if either the dimension of the building is larger than , if is not isomorphic to for some prime , or if the building is associated to , where is a noncommutative division algebra. Along the way we also give a new proof of the Siegel–Klingen theorem on the rationality of certain Dedekind zeta functions and -functions.