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For the $d$-dimensional incompressible Euler equation, the usual energy method gives local well-posedness for initial velocity in Sobolev space $H^s(\mathbb{R}^d)$, $s>s_c:=d/2+1$. The borderline case $s=s_c$ was a folklore conjecture. In the previous paper [2], we introduced a new strategy (large lagrangian deformation and high frequency perturbation) and proved strong ill-posedness in the critical space $H^1(\mathbb{R}^2)$. The main issues in 3D are vorticity stretching, lack of $L^p$ conservation, and control of lifespan. Nevertheless in this work we overcome these difficulties and show strong ill-posedness in 3D. Our results include general borderline Sobolev and Besov spaces.

By using an idea of localized Galilean boost, we show that the data-to-solution map for incompressible Euler equations is not uniformly continuous in $${H^s({\mathbb{R}}^d)}$$, $${s \ge 0}$$. This settles the end-point case (s = 0) left open in Himonas–Misiolek (Commun Math Phys 296(1):285–301, 2010) and gives a unified treatment for all Hs. We also show the solution map is nowhere uniformly continuous.

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., Comment: Some notation has been revised, and the referee's suggestions addressed. The result now covers a larger class of interaction potentials. To appear in Geometric and Functional Analysis

This is a survey article about recent developments in dimension-free estimates for maximal functions corresponding to the Hardy–Littlewood averaging operators associated with convex symmetric bodies in \(\mathbb R^d\) and \(\mathbb Z^d\).

We show that the discrete Hardy–Littlewood maximal functions associated with the Euclidean balls in \(\mathbb Z^d\) with dyadic radii have bounds independent of the dimension on \(\ell ^p(\mathbb Z^d)\) for p ∈ [2, ∞].

We present three applications of the Siegel mass formula, using the explicit upper bounds for densities derived in Bourgain and Demeter (Int Math Res Not 2015(11):3150–3184, 2014).

We extend the decoupling results of the first two authors to the case of real analytic surfaces of revolution in \({\mathbb R}^3\). New examples of interest include the torus and the perturbed cone.

Propositions 1.1–1.3 stated below contribute to results and certain problems considered in [E-S], on the behavior of products $$\Pi^n_1(1-z^{a_j}),1\leq{a_1}...\leq{a_n}$$ integers. In the discussion below, {a1,..., an} will be either a proportional subset of {1,..., n} or a set of large arithmetic diameter.

We study dimension-free L p inequalities for r-variations of the Hardy–Littlewood averaging operators defined over symmetric convex bodies in $${\mathbb{R}^d}$$ .

We prove a sharp decoupling for a certain two dimensional surface in R 9 . As an application, we obtain the full range of expected estimates for the cubic Parsell–Vinogradov system in two dimensions.

The purpose of this paper is to present some further applications of the general decoupling theory from [B-D] and [B-D2] to certain diophantine issues. In particular, we consider 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.

We obtain smoothing estimates for certain nonlinear convolution operators on prime fields, leading to quantitative nonlinear Roth type theorems. Compared with the usual linear setting (i.e. arithmetic progressions), the nonlinear nature of the operators leads to different phenomena, both qualitatively and quantitatively.

We prove that a uniformly bounded system of orthonormal functions satisfying the $\psi_2$ condition: (1) must contain a Sidon subsystem of proportional size, (2) must satisfy the Rademacher-Sidon property, and (3) must have its 5-fold tensor satisfy the Sidon property. On the other hand, we construct a uniformly bounded orthonormal system that satisfies the $\psi_2$ condition but which is not Sidon. These problems are variants of Kaczmarz's Scottish book problem (problem 130) which, in its original formulation, was answered negatively by Rudin. A corollary of our argument is a new, elementary proof of Pisier's theorem that a set of characters satisfying the $\psi_2$ condition is Sidon.

This paper contains a detailed, self contained and more streamlined proof of the l2 decoupling theorem for hypersurfaces from the paper of Bourgain and Demeter in 2015. The authors hope this will serve as a good warm up for the readers interested in understanding the proof of Vinogradov’s mean value theorem from the paper of Bourgain, Demeter and Guth in 2015.

It is shown that control of the Schrodinger maximal functions $\sup_{0, Comment: 4 pages

We use decoupling theory to estimate the number of solutions for quadratic and cubic Parsell--Vinogradov systems in two dimensions., Final version, to appear in JLMS. arXiv admin note: text overlap with arXiv:1508.01149

We establish a quantitative version of Oppenheim's conjecture for generic ternary indefinite quadratic forms using an analytic number theory approach. The statements come with power gains and in some cases are essentially optimal, Comment: 8 pages

We prove a sharp decoupling for nondegenerate surfaces in R 4 . This puts the progress in [1] on the Lindelof hypothesis into a more general perspective.

We investigate the transitivity properties of the group of morphisms generated by Vieta involutions on the solutions in congruences to the Markoff equation as well as to other Markoff type affine cubic surfaces. These are dictated by the finite Q ¯ orbits of these actions and these can be determined effectively. The results are applied to give forms of strong approximation for integer points, and to sieving, on these surfaces.

We study the Fourier–Walsh spectrum { µ(S);S ⊂{ 1,...,n}} of the Moebius function µ restricted to {0,1,2,...,2n −1 }�{ 0,1}n and prove that it is not captured by levels { µ(S) || S

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, 2 figures. Included reviewer comments. Accepted for the Contemporary Mathematics proceedings of Constructive Functions 2014

The main result of the paper is the construction of explicit uniformly bounded bases 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.

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 $\delta$ of the limit set, in particular we do not require the pressure condition $\delta\leq {1\over 2}$. 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 $\delta$-regular sets with $\delta, Comment: 39 pages, 5 figures. Added explanations of the proof (especially for Theorem 4) and revised according to referee's comments. To appear in Ann. Math

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 $p$.

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 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 L 1 norm of the restriction of the eigenfunction to the curve, and then to an upper bound for the L 4 restriction norm.

We consider the d-dimensional incompressible Euler equations. We show strong illposedness of velocity in any Cm spaces whenever m ≥ 1 is an integer. More precisely, we show for a set of initial data dense in the Cm topology, the corresponding solutions lose Cm regularity instantaneously in time. In the C1 case, our proof is based on an anisotropic Lagrangian deformation and a short-time flow expansion. In the Cm, m ≥ 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 Cm−1,1 whenever m ≥ 1 is an integer.

We study an instance of the Affine Sieve, producing a level of distribution beyond that which can be obtained from current techniques, even assuming a Selberg/Ramanujan-type spectral gap. In particular, we consider the set of hypotenuses in a thin orbit of Pythagorean triples. Previous work [Kon07, Kon09, KO12] gave an exponent of distribution alpha < 1/12 coming from Gamburd's [Gam02] gap theta = 5/6, thereby producing R = 13 almost primes in this linear sieve problem (see Sec. 1 for definitions). If conditioned on a best possible gap theta = 1/2, the known method would give an exponent alpha < 1/4, and R = 5 almost primes. The exponent 1/4 is the natural analogue of the "Bombieri-Vinogradov" range of distribution for this problem, see Remark 1.19. In this paper, we unconditionally prove the exponent alpha < 7/24 (in the "Elliott-Halberstam" range), thereby producing R = 4 almost primes. The main tools involve developing bilinear forms and the dispersion method in the range of incomplete sums for this Affine Sieve problem.

We settle a question of Farrell and Vershynin on the inverse of the perturbation of a given arbitrary symmetric matrix by a GOE element.

We obtain an essential spectral gap for a convex co-compact hyperbolic surface $M=\Gamma\backslash\mathbb H^2$ which depends only on the dimension $\delta$ of the limit set. More precisely, we show that when $\delta>0$ there exists $\varepsilon_0=\varepsilon_0(\delta)>0$ such that the Selberg zeta function has only finitely many zeroes $s$ with $\Re s>\delta-\varepsilon_0$. 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 $\Gamma$ are nonlinear, together with estimates on exponential sums due to Bourgain which follow from the discretized sum-product theorem in $\mathbb R$., Comment: 28 pages, 4 figures. Added Figure 1 on page 2 and made other minor changes. To appear in GAFA

Let G be the group SO(d) or SU(d) with d large. How long does it take for a random walk on G to approximate uniform measure? It is shown that in certain natural examples an ɛ-approximation is achieved in time \(\left (d\log \frac{1} {\varepsilon } \right )^{C}\).

We study a homogenization question for stochastic divergence type operator, Comment: 8 pages

For the $d$-dimensional incompressible Euler equation, the standard energy method gives local wellposedness for initial velocity in Sobolev space $H^s(\mathbb R^d)$, $s>s_c:=d/2+1$. The borderline case $s=s_c$ was a folklore open problem. In this paper we consider the physical dimensions $d=2,3$ and show that if we perturb any given smooth initial data in $H^{s_c}$ norm, then the corresponding solution can have infinite $H^{s_c}$ norm instantaneously at $t>0$. The constructed solutions are unique and even $C^{\infty}$-smooth in some cases. To prove these results we introduce a new strategy: large Lagrangian deformation induces critical norm inflation. As an application we also settle several closely related open problems., 119 pages

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 $p>\frac 32$., 20 pages

Recently, several bounds have been obtained on the number of solutions of congruences of the type $$({x_1} + s) \cdots ({x_v} + s) \equiv ({y_1} + s) \cdots ({y_v} + s)\not \equiv 0{\text{ (mod }}p{\text{),}}$$ where p is prime and variables take values in some short interval. 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 p, one can always find an element of a large order in any rather short interval.

We obtain new results on the additive properties of the set , where is an arbitrary interval in the field of residue classes modulo a large prime . Combining our results with estimates of multilinear exponential sums, we obtain new results on incomplete multilinear Kloosterman sums.

The study of sum and product problems in finite fields motivates the investigation of additive structures in multiplicative subgroups of such fields. A simple known fact is that any multiplicative subgroup of size at least q 3/4 in the finite field F q must contain an additive relation x + y = z. Our main result is that there are infinitely many examples of sum-free multiplicative subgroups of size Ω(p 1/3) in prime fields F p . More complicated additive relations are studied as well. One representative result is the fact that the elements of any multiplicative subgroup H of size at least q 3/4+o(1) of F q can be arranged in a cyclic permutation so that the sum of any pair of consecutive elements in the permutation belongs to H. The proofs combine combinatorial techniques based on the spectral properties of Cayley sum-graphs with tools from algebraic and analytic number theory.

We establish new global well-posedness results along Gibbs measure evolution for the nonlinear wave equation posed on the unit ball in $\mathbb{R}^3$ via two distinct approaches. The first approach invokes the method established in the works \cite{B1,B2,B3} based on a contraction-mapping principle and applies to a certain range of nonlinearities. The second approach allows to cover the full range of nonlinearities admissible to treatment by Gibbs measure, working instead with a delicate analysis of convergence properties of solutions. The method of the second approach is quite general, and we shall give applications to the nonlinear Schr\"odinger equation on the unit ball in subsequent works \cite{BB1,BB2}., Comment: 22 pages