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Jean Bourgain

Publications and source records attributed to Jean Bourgain.

At least 19 recordsLinked to original sources
arXiv2023

On a multi-parameter variant of the Bellow-Furstenberg problem

We prove convergence in norm and pointwise almost everywhere on $L^p$, $p\in (1,\infty)$, for certain multi-parameter polynomial ergodic averages by establishing the corresponding multi-parameter maximal and oscillation inequalities. Our result, in particular, gives an affirmative answer to a multi-parameter variant of the Bellow-Furstenberg problem. This paper is also the first systematic treatment of multi-parameter oscillation semi-norms which allows an efficient handling of multi-parameter pointwise convergence problems with arithmetic features. The methods of proof of our main result develop estimates for multi-parameter exponential sums, as well as introduce new ideas from the so-called multi-parameter circle method in the context of the geometry of backwards Newton diagrams that are dictated by the shape of the polynomials defining our ergodic averages.

math.DS↗
arXiv2023

Mean square of zeta function, circle problem and divisor problem revisited

This paper is closely related to the recent work [BW17] of the same authors and our purpose is to elaborate more on some of the results and methods from [BW17]. More specifically our goal is two-fold. Firstly, we will indicate how a simple variant related to Section 4 in [BW17] leads to the following improvements of Theorem 3 in [BW17]

math.AP↗
arXiv2019

Three applications of the Siegel mass formula

We present three applications of the Siegel mass formula. First we estimate the number of solutions of a quadratic system of equations. We also include estimates for the distribution of lattice points on caps of four dimensional spheres and for the number of non-congruent lattice tetrahedra.

math.NT↗
arXiv2019

Dimension-free estimates for discrete Hardy-Littlewood averaging operators over the cubes in $\mathbb Z^d$

Dimension-free bounds will be provided in maximal and $r$-variational inequalities on $\ell^p(\mathbb Z^d)$ corresponding to the discrete Hardy-Littlewood averaging operators defined over the cubes in $\mathbb Z^d$. We will also construct an example of a symmetric convex body in $\mathbb Z^d$ for which maximal dimension-free bounds fail on $\ell^p(\mathbb Z^d)$ for all $p\in(1, \infty)$. Finally, some applications in ergodic theory will be discussed.

math.CA↗
arXiv2018

Anderson localization for two interacting quasiperiodic particles

We consider a system of two discrete quasiperiodic 1D particles as an operator on $\ell^2(\mathbb Z^2)$ and establish Anderson localization at large disorder, assuming the potential has no cosine-type symmetries. In the presence of symmetries, we show localization outside of a neighborhood of finitely many energies. One can also add a deterministic background potential of low complexity, which includes periodic backgrounds and finite range interaction potentials. Such background potentials can only take finitely many values, and the excluded energies in the symmetric case are associated to those values.

math.SP↗
arXiv2018

Beyond Expansion IV: Traces of Thin Semigroups

We continue our study of particular instances of the Affine Sieve, producing levels of distribution beyond those attainable from expansion alone. Motivated by McMullen's Arithmetic Chaos Conjecture regarding low-lying closed geodesics on the modular surface defined over a given number field, we study the set of traces for certain sub-semi-groups of SL2(Z) corresponding to absolutely Diophantine numbers. In particular, we are concerned with the level of distribution for this set. While the standard Affine Sieve procedure, combined with Bourgain-Gamburd-Sarnak's resonance-free region for the resolvent of a "congruence" transfer operator, produces some exponent of distribution alpha > 0, we are able to produce the exponent alpha < 1/3. This recovers unconditionally the same exponent as what one would obtain under a Ramanujan-type conjecture for thin groups. A key ingredient, of independent interest, is a bound on the additive energy of SL2(Z).

math.NT↗
arXiv2017

Spectral gaps without the pressure condition

For all convex co-compact hyperbolic surfaces, we prove the existence of an essential spectral gap, that is a strip beyond the unitarity axis in which the Selberg zeta function has only finitely many zeroes. We make no assumption on the dimension $δ$ of the limit set, in particular we do not require the pressure condition $δ\leq {1\over 2}$. This is the first result of this kind for quantum Hamiltonians. Our proof follows the strategy developed by Dyatlov-Zahl [arXiv:1504.06589]. The main new ingredient is the fractal uncertainty principle for $δ$-regular sets with $δ<1$, which may be of independent interest.

math.CA↗
arXiv2017

Fourier dimension and spectral gaps for hyperbolic surfaces

We obtain an essential spectral gap for a convex co-compact hyperbolic surface $M=Γ\backslash\mathbb H^2$ which depends only on the dimension $δ$ of the limit set. More precisely, we show that when $δ>0$ there exists $\varepsilon_0=\varepsilon_0(δ)>0$ such that the Selberg zeta function has only finitely many zeroes $s$ with $\Re s>δ-\varepsilon_0$. The proof uses the fractal uncertainty principle approach developed by Dyatlov-Zahl [arXiv:1504.06589]. The key new component is a Fourier decay bound for the Patterson-Sullivan measure, which may be of independent interest. This bound uses the fact that transformations in the group $Γ$ are nonlinear, together with estimates on exponential sums due to Bourgain which follow from the discretized sum-product theorem in $\mathbb R$.

math.CA↗
arXiv2016

Beyond Expansion III: Reciprocal Geodesics

We prove the existence of infinitely many low-lying and fundamental closed geodesics on the modular surface which are reciprocal, that is, invariant under time reversal. The method combines ideas from Parts I and II of this series, namely the dispersion method in bilinear forms, as applied to thin semigroups coming from restricted continued fractions.

math.NT↗
arXiv2016

Small gaps in the spectrum of the rectangular billiard

We study the size of the minimal gap between the first N eigenvalues of the Laplacian on a rectangular billiard having irrational squared aspect ratio $α$, in comparison to the corresponding quantity for a Poissonian sequence. If $α$ is a quadratic irrationality of certain type, such as the square root of a rational number, we show that the minimal gap is roughly of size 1/N, which is essentially consistent with Poisson statistics. We also give related results for a set of $α$'s of full measure. However, on a fine scale we show that Poisson statistics is violated for all $α$. The proofs use a variety of ideas of an arithmetical nature, involving Diophantine approximation, the theory of continued fractions, and results in analytic number theory.

math.AP↗