Optimal mean value estimates for the quadratic Weyl sums in two dimensions
We use decoupling theory to prove a sharp (up to $N^ε$ losses) estimate for Vinogradov's mean value theorem in two dimensions
math.CA↗
arXiv subjects
Jean Bourgain
Publications and source records attributed to Jean Bourgain.
Thermodynamic expansion to arbitrary moduli
We extend the thermodynamic expansion results in [BourgainGamburdSarnak2011, MageeOhWinter2015] from square-free to arbitrary moduli by developing a novel decoupling technique and applying [BourgainVarju2012].
math.NT↗
The proof of the $l^2$ Decoupling Conjecture
We prove the $l^2$ Decoupling Conjecture for compact hypersurfaces with positive definite second fundamental form and also for the cone. This has a wide range of important consequences. One of them is the validity of the Discrete Restriction Conjecture, which (up to $N^ε$ losses) implies the full range of expected $L^p_{x,t}$ Strichartz estimates for both classical and irrational tori. Another one is an improvement in the range for the discrete restriction theory for lattice points on the sphere. Various applications in Additive Combinatorics, Incidence Geometry and Number Theory are also discussed. Our argument relies on the interplay between linear and multilinear restriction theory.
math.CA↗
On uniformly bounded basis in spaces of holomorphic functions
The main result of the paper is the construction of explicit uniformly bounded basis in the spaces of complex homogenous polynomials on the unit ball of $C^3$, extending an earlier result of the author in the $C^2$ case
math.FA↗
On Random Walks in large compact Lie groups
We evaluate the mixing time of certain random walks on large unitary groups
math.PR↗
Decoupling for perturbed cones and mean square of $|ζ(\frac 12+it)|$
An improved estimate is obtained for the mean square of the modulus of the zeta function on the critical line. It is based on the decoupling techniques in harmonic analysis developed in [B-D]
math.NT↗
Local statistics of lattice points on the sphere
A celebrated result of Legendre and Gauss determines which integers can be represented as a sum of three squares, and for those it is typically the case that there are many ways of doing so. These different representations give collections of points on the unit sphere, and a fundamental result, conjectured by Linnik, is that under a simple condition these become uniformly distributed on the sphere. In this note we survey some of our recent work, which explores what happens beyond uniform distribution, giving evidence to randomness on smaller scales. We treat the electrostatic energy, local statistics such as the point pair statistic (Ripley's function), nearest neighbour statistics, minimum spacing and covering radius. We briefly discuss the situation in other dimensions, which is very different. In an appendix we compute the corresponding quantities for random points
math.NT↗
Decouplings for surfaces in R^4
We prove a sharp decoupling for non degenerate surfaces in $\R^4$. This puts the recent progress on the Lindelöf hypothesis into a more general perspective.
math.CA↗
Strong illposedness of the incompressible Euler equation in integer $C^m$ spaces
We consider the $d$-dimensional incompressible Euler equations. We show strong illposedness of velocity in any $C^m$ spaces whenever $m\ge 1$ is an \emph{integer}. More precisely, we show for a set of initial data dense in the $C^m$ topology, the corresponding solutions lose $C^m$ regularity instantaneously in time. In the $C^1$ case, our proof is based on an anisotropic Lagrangian deformation and a short-time flow expansion. In the $C^m$, $m\ge 2$ case, we introduce a flow decoupling method which allows to tame the nonlinear flow almost as a passive transport. The proofs also cover illposedness in Lipschitz spaces $C^{m-1,1}$ whenever $m\ge 1$ is an integer.
math.AP↗
Influential coalitions for Boolean Functions
We improve results of Kahn, Kalai, and Linial from the late 80s on the existence of influential large coalitions for Boolean functions, and we give counterexamples to conjectures (of Benny Chor and others) also from the late 80s, by exhibiting functions for which the influences of large coalitions are unexpectedly small relative to the expectations of the functions. The large gaps between the new upper and lower bounds leave a lot of room for further study.
math.CO↗
BMO-type norms related to the perimeter of sets
In this paper we consider an isotropic variant of the $BMO$-type norm recently introduced by Bourgain, Brezis and Mironescu. We prove that, when considering characteristic functions of sets, this norm is related to the perimeter. A byproduct of our analysis is a new characterization of the perimeter of sets in terms of this norm, independent of the theory of distributions.
math.FA↗
$l^p$ decouplings for hypersurfaces with nonzero Gaussian curvature
We prove an $l^p$ decoupling inequality for hypersurfaces with nonzero Gaussian curvature and use it to derive a corresponding $l^p$ decoupling for curves not contained in a hyperplane. This extends our earlier work from [2]
math.CA↗
Decoupling inequalities and some mean-value theorems
The purpose of this paper is to present some further applications of the general decoupling theory from [B-D1, 2] to certain diophantine issues. In particular, we concider mean value estimates relevant to the Bombieri-Iwaniec approach to exponential sums and arising in the work of Robert and Sargos [R-S]. Our main input is a new mean value theorem.
math.NT↗
A Finitary Hasse Principle for Diagonal Curves
We prove a Hasse principle for solving equations of the form ax+by+cz=0 where x, y, z belong to a given finite index subgroup of the multiplicative group of rational numbers. From this we deduce a Hasse principle for diagonal curves over fields of algebraic numbers with finitely generated Galois group.
math.NT↗
Character Sums and Deterministic Polynomial Root Finding in Finite Fields
We obtain a new bound of certain double multiplicative character sums. We use this bound together with some other previously obtained results to obtain new algorithms for finding roots of polynomials modulo a prime $p$.
math.NT↗
Nodal intersections and Lp restriction theorems on the torus
We study the number of intersections of the nodal lines of an eigenfunction of the Laplacian on the standard torus with a fixed reference curve, that is, the number of zeros of the eigenfunction restricted to the curve. An upper bound is the wave number k. When the curve has nowhere zero curvature, we conjecture that, up to a constant multiple, this should also be the correct a lower bound. We give a lower bound which differs from this by an arithmetic quantity, given in terms of the maximal number of lattice points in arcs of size square root of the wave number k on a circle of radius k. According to a conjecture of Cilleruelo and Granville, this quantity is bounded in which case we recover our conjecture. To get at the lower bound, we reduce the problem to giving a lower bound for the L1 norm of the restriction of the eigenfunction to the curve, and then to an upper bound for the L4 restriction norm.
math.AP↗
Affine extractors over large fields with exponential error
We describe a construction of explicit affine extractors over large finite fields with exponentially small error and linear output length. Our construction relies on a deep theorem of Deligne giving tight estimates for exponential sums over smooth varieties in high dimensions.
cs.CC↗
A Note on Lyapunov Exponents of Deterministic Strongly Mixing Potentials
In this Note, we consider 1D lattice Schrodinger operators with deterministic strongly mixing potentials with very small coupling. We describe a scheme to establish positiv- ity of the Lyapunov exponent from a statement at some fixed scale. The required input may then be derived from Furstenberg theory, if the underlying dynamics are sufficiently mixing, or verified directly by numerical means.
math-ph↗