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This monograph is devoted to the proof of two related results. The first one asserts that if The second result of the monograph deals with the relationship between the above square function in the complex plane and the Cauchy transform

We extend the proof of Reifenberg’s Topological Disk Theorem to allow the case of sets with holes, and give sufficient conditions on a set On généralise la démonstration du théorème du disque topologique de Reifenberg pour inclure le cas d’ensembles ayant des trous, et on donne des conditions suffisantes sur l’ensemble

In this paper we show that generic continuous Lebesgue measure-preserving circle maps have the s-limit shadowing property. In addition, we obtain that s-limit shadowing is a generic property also for continuous circle maps. In particular, this implies that classical shadowing, periodic shadowing and limit shadowing are generic in these two settings as well.

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 ≤ 1/2. This is the first result of this kind for quantum Hamiltonians. Our proof follows the strategy developed by Dyatlov and Zahl. The main new ingredient is the fractal uncertainty principle for δ-regular sets with δ < 1, which may be of independent interest.

We consider a large class of 2D area-preserving diffeomorphisms that are not uniformly hyperbolic but have strong hyperbolicity properties on large regions of their phase spaces. A prime example is the standard map. Lower bounds for Lyapunov exponents of such systems are very hard to estimate, due to the potential switching of \"stable\" and \"unstable\" directions. This paper shows that with the addition of (very) small random perturbations, one obtains with relative ease Lyapunov exponents reflecting the geometry of the deterministic maps.

We derive an asymptotic expansion for the excess risk (regret) of a weighted nearest-neighbour classifier. This allows us to find the asymptotically optimal vector of nonnegative weights, which has a rather simple form. We show that the ratio of the regret of this classifier to that of an unweighted k-nearest neighbour classifier depends asymptotically only on the dimension d of the feature vectors, and not on the underlying populations. The improvement is greatest when d = 4, but thereafter decreases as d → ∞. The popular bagged nearest neighbour classifier can also be regarded as a weighted nearest neighbour classifier, and we show that its corresponding weights are somewhat suboptimal when d is small (in particular, worse than those of the unweighted k-nearest neighbour classifier when d = 1), but are close to optimal when d is large. Finally, we argue that improvements in the rate of convergence are possible under stronger smoothness assumptions, provided we allow negative weights. Our findings are supported by an empirical performance comparison on both simulated and real data sets.

A thin set is defined to be an uncountable dense zero-dimensional subset of measure zero and Hausdorff measure zero of an Euclidean space. A fat set is defined to be an uncountable dense path-connected subset of an Euclidean space which has full measure, that is, its complement has measure zero. While there are well-known pathological maps of a set of measure zero, such as the Cantor set, onto an interval, we show that the standard addition on $\\mathbb {R}$ maps a thin set onto a fat set; in fact the fat set is all of $\\mathbb {R}$ . Our argument depends on the theorem of Paul Erdős that every real number is a sum of two Liouville numbers. Our thin set is the set $\\mathcal {L}^{2}$ , where $\\mathcal {L}$ is the set of all Liouville numbers, and the fat set is $\\mathbb {R}$ itself. Finally, it is shown that $\\mathcal {L}$ and $\\mathcal {L}^{2}$ are both homeomorphic to $\\mathbb {P}$ , the space of all irrational numbers.

During a first or second course in number theory, students soon encounter several sets of \"number theoretic interest\". These include basic sets such as the rational numbers, algebraic numbers, transcendental numbers, and Liouville numbers, as well as more exotic sets such as the constructible numbers, normal numbers, computable numbers, badly approximable numbers, the Mahler sets S, T and U, and sets of irrationality exponent m , among others. Those exposed to some measure theory soon make a curious observation regarding a common property seemingly shared by all these sets: each of the sets has Lebesgue measure equal to zero, or its complement has Lebesgue measure equal to zero. In this expository note, we explain this phenomenon.

We consider the d-dimensional nonlinear Schrödinger equation under periodic boundary conditions: $-i\\dot{u}=-\\Delta u+V(x)\\ast u+\\varepsilon \\frac{\\partial F}{\\partial \\overline{u}}(x,u,\\overline{u}),\\quad u=u(t,x),x\\in {\\Bbb T}^{d}$ where $V(x)=\\sum \\hat{V}(a)e^{i\\langle a,x\\rangle}$ is an analytic function with V̂ real, and F is a real analytic function in 𝓡u, 𝓣u and x. (This equation is a popular model for the 'real' NLS equation, where instead of the convolution term V * u we have the potential term Vu.) For ε = 0 the equation is linear and has time—quasi-periodic solutions $u(t,x)=\\sum_{a\\in {\\cal A}}\\hat{u}(a)e^{i(|a|^{2}+\\hat{V}(a))t}e^{i\\langle a,x\\rangle},\\quad|\\hat{u}(a)|>0,$ where 𝒜 is any finite subset of ℤ d . We shall treat ω a = ǀaǀ² + V̂ (a), a ∈ 𝒜, as free parameters in some domain 𝑈⊂ℝ 𝒜 . This is a Hamiltonian system in infinite degrees of freedom, degenerate but with external parameters, and we shall describe a KAM-theory which, under general conditions, will have the following consequence: If ǀεǀ is sufficiently small, then there is a large subset U′ of U such that for all ω ∈ U′ the solution u persists as a time—quasi-periodic solution which has all Lyapounov exponents equal to zero and whose linearized equation is reducible to constant coefficients.

In 1939 Turán raised the question about lower estimations of the maximum norm of the derivatives of a polynomial p of maximum norm 1 on the compact set K of the complex plain under the normalization condition that the zeroes of p in question all lie in K . Turán studied the problem for the interval I and the unit disk D and found that with n : = deg p tending to infinity, the precise growth order of the minimal possible derivative norm (oscillation order) is n for I and n for D . Erőd continued the work of Turán considering other domains. Finally, in 2006, Halász and Révész proved that the growth of the minimal possible maximal norm of the derivative is of order n for all compact convex domains. Although Turán himself gave comments about the above oscillation question in L q norms, till recently results were known only for D and I . Recently, we have found order n lower estimations for several general classes of compact convex domains, and proved that in L q norm the oscillation order is at least n / log n for all compact convex domains. In the present paper we prove that the oscillation order is not greater than n for all compact (not necessarily convex) domains K and L q norm with respect to any measure supported on more than two points on K .