A keystone in the classical theory of diophantine approximation is the construction of an auxilliary polynomial. The polynomial is constructed so that it is forced (for arithmetic reasons) to vanish at certain approximating points and this contradicts an upper bound on the order of vanishing obtained by other (usually geometric) techniques; the contradiction then allows one to prove finiteness ...

As every vertex of this graph has one “outgoing” and at most one “incoming” edge, each of those components is a cycle or an infinite path. In each of these paths and cycles we now select every other edge to mark the desired bijection. The Cantor-Bernstein problem, rephrased as above for graphs, has a natural generalization to paths. Let G be any graph, and let A and B be disjoint sets of vertic...

Abstract We investigate stable intersections of conformal Cantor sets and their consequences to dynamical systems. First we define this type set relate it horseshoes appearing in automorphisms $\mathbb {C}^2$ . Then study limit geometries, that is, objects related the asymptotic shape sets, obtain a criterion guarantees intersection between some configurations. Finally, show Buzzard constructio...

We prove $L^p$ bounds for the maximal operators associated to an Ahlfors-regular variant of fractal percolation. Our improve upon those obtained by I. Łaba and M. Pramanik in some cases are sharp up endpoint. A consequence our main result is that there exist Salem Cantor sets any dimension $>1/2$ such operator bounded on $L^2(\mathbb{R})$. follow overall scheme Łaba-Pramanik analytic part argum...

This paper is concerned with discrete, one-dimensional Schrödinger operators with real analytic potentials and one Diophantine frequency. Using localization and duality we show that almost every point in the spectrum admits a quasi-periodic Bloch wave if the potential is smaller than a certain constant which does not depend on the precise Diophantine conditions. The associated first-order syste...

A classical theorem of Fekete and Szegő [4] says that if E is a compact set in the complex plane, stable under complex conjugation and having logarithmic capacity γ(E) ≥ 1, then every neighborhood of E contains infinitely many conjugate sets of algebraic integers. Raphael Robinson [5] refined this, showing that if E is contained in the real line, then every neighborhood of E contains infinitely...

Ordinals carry a natural topology induced by their linear order. In this note, we classify the homeomorphism types of all ordinal topologies using the Cantor normal form and the notion of the Cantor–Bendixson rank of a point. Let < be a linear order on a set X. The order topology on X is generated by the subbase of rays of the form leftX(b) := {x ∈ X : x < b} and rightX(a) := {x ∈ X : a < x} fo...

Erdos and Rado defined a A-system, as a family in which every two members have the same intersection. Here we obtain a new upper bound on the maximum cardinality q ( n , q ) of an n-uniform family not containing any A-system of cardinality q. Namely, we prove that, for any a > 1 and q , there exists C = C(a, q ) such that, for any n ,

The compact Hausdorff space X has the Complex Stone-Weierstrass Property (CSWP) iff it satisfies the complex version of the Stone-Weierstrass Theorem. W. Rudin showed that all scattered spaces have the CSWP. We describe some techniques for proving that certain non-scattered spaces have the CSWP. In particular, if X is the product of a compact ordered space and a compact scattered space, then X ...