arXiv subjects
Publications and source records attributed to Jean Bourgain.
The purpose of this Note is to provide a deterministic implementation of the random wave model for the number of nodal domains in the context of the two-dimensional torus. The approach is based on recent work due to Nazarov and Sodin and arithmetical properties of lattice points on circles.
We extend the convergence method introduced in our works [8]-[10] for almost sure global well-posedness of Gibbs measure evolutions of the nonlinear Schrödinger (NLS) and nonlinear wave (NLW) equations on the unit ball in R^d to the case of the three dimensional NLS. This is the first probabilistic global well-posedness result for NLS with supercritical data on the unit ball in R^3. The initial data is taken as a Gaussian random process lying in the support of the Gibbs measure associated to the equation, and results are obtained almost surely with respect to this probability measure. The key tools used include a class of probabilistic a priori bounds for finite-dimensional projections of the equation and a delicate trilinear estimate on the nonlinearity, which - when combined with the invariance of the Gibbs measure - enables the a priori bounds to be enhanced to obtain convergence of the sequence of approximate solutions.
For the Schrödinger equation, $ (i \partial_t + Δ) u = 0 $ on a torus, an arbitrary non-empty open set $ Ω$ provides control and observability of the solution: $ \| u |_{t = 0} \|_{L^2 (\T^2)} \leq K_T \| u \|_{L^2 ([0,T] \times Ω)} $. We show that the same result remains true for $ (i \partial_t + Δ- V) u = 0 $ where $ V \in L^2 (\T^2) $, and $ \T^2 $ is a (rational or irrational) torus. That extends the results of \cite{AM}, and \cite{BZ4} where the observability was proved for $ V \in C (\T^2) $ and conjectured for $ V \in L^\infty (\T^2) $. The higher dimensional generalization remains open for $ V \in L^\infty (\T^n) $.
It is shown that the Hardy-Littlewood maximal function associated to the cube in $\mathbb R^n$ obeys dimensional free bounds in $L^p$ fir $p>1$. Earlier work only covered the range $p>\frac 32$.
It is shown that monotone Boolean functions on the Boolean cube capture the expected number of primes, under he usual identification by binary expansion. This answers a question posed by G.Kalai.
In the paper, the occurrence of zeros and ones in the binary expansion of the primes is studied. In particular the statement in the title is established. The proof is unconditional.
Recently, several bounds have been obtained on the number of solutions to congruences of the type $$ (x_1+s)...(x_ν+s)\equiv (y_1+s)...(y_ν+s)\not\equiv0 \pmod p $$ modulo a prime $p$ with variables from some short intervals. Here, for almost all $p$ and all $s$ and also for a fixed $p$ and almost all $s$, we derive stronger bounds. We also use similar ideas to show that for almost all primes, one can always find an element of a large order in any rather short interval.
We establish bounds on the density of states measure for Schrödinger operators. These are deterministic results that do not require the existence of the density of states measure, or, equivalently, of the integrated density of states. The results are stated in terms of a "density of states outer-measure" that always exists, and provides an upper bound for the density of states measure when it exists. We prove log-Hölder continuity for this density of states outer-measure in one, two, and three dimensions for Schrödinger operators, and in any dimension for discrete Schrödinger operators.
We establish a Szemeredi-Trotter type result for hyperbolas in $\mathbb F_p\times\mathbb F_p$
It is shown that there is an absolute constant $C$ such that any rational $\frac bq\in]0, 1[, (b, q)=1$, admits a representation as a finite sum $\frac bq=\sum_α\frac {b_α}{q_α}$ where $\sum_α\sum_ia_i(\frac {b_α}{q_α})<C\log q$ and $\{a_i(x)\}$ denotes the sequence of partial quotients of $x$.
We establish new results for the radial nonlinear wave and Schrödinger equations on the ball in $\Bbb R^2$ and $\Bbb R^3$, for random initial data. More precisely, a well-defined and unique dynamics is obtained on the support of the corresponding Gibbs measure. This complements results from \cite{B-T1,B-T2} and \cite {T1,T2}.
We obtain upper bounds on the number of solutions to congruences of the type $$ (x_1+s)...(x_ν+s)\equiv (y_1+s)...(y_ν+s)\not\equiv0 \pmod p $$ modulo a prime $p$ with variables from some short intervals. We give some applications of our results and in particular improve several recent estimates of J. Cilleruelo and M. Z. Garaev on exponential congruences and on cardinalities of products of short intervals, some double character sum estimates of J. B. Friedlander and H. Iwaniec and some results of M.-C. Chang and A. A. Karatsuba on character sums twisted with the divisor function.
It is shown that Schrodinger operators defined from the standard map have positive (mean) Lyapounov exponents for almost all energies
New estimates on the maximal function associated to the linear Schrodinger equation are established
It is shown that any primitive integral Apollonian circle packing captures a fraction of the prime numbers. Basically the method consists in applying the circle method, considering the curvatures produced by a well-chosen family of binary quadratic forms.
We present a new result on counting primes $p<N=2^n$ for which $r$ (arbitrarily placed) digits in the binary expansion of $p$ are specified. Compared with earlier work of Harman and Katai, the restriction on $r$ is relaxed to $r< c\Big(\frac n{\log n}\Big)^{4/7}$. This condition results from the estimates of Gallagher and Iwaniec on zero-free regions of $L$-functions with `powerful' conductor.
We consider the problem of recovering a hidden element $s$ of a finite field $\F_q$ of $q$ elements from queries to an oracle that for a given $x\in \F_q$ returns $(x+s)^e$ for a given divisor $e\mid q-1$. We use some techniques from additive combinatorics and analytic number theory that lead to more efficient algorithms than the naive interpolation algorithm, for example, they use substantially fewer queries to the oracle.
We study the Fourier-Walsh spectrum $\{\hatμ(S); S\subset\{1, ..., n\}\}$ of the Moebius function $μ$ restricted to $\{0, 1, 2, ..., 2^n-1\}\simeq \{0, 1\}^n$ and prove that it is not captued by levels \{\hatμ(S)| \, |S|< n^{\frac 23-\epsion}\}. An application to correlation with monotone Boelean functions is given.