563 research outputs found
Intermediate convergents and a metric theorem of Khinchin
A landmark theorem in the metric theory of continued fractions begins this
way: Select a non-negative real function defined on the positive integers
and a real number , and form the partial sums of evaluated at the
partial quotients in the continued fraction expansion for .
Does the sequence have a limit as n\rar\infty? In 1935 A. Y.
Khinchin proved that the answer is yes for almost every , provided that the
function does not grow too quickly. In this paper we are going to explore a
natural reformulation of this problem in which the function is defined on
the rationals and the partial sums in question are over the intermediate
convergents to with denominators less than a prescribed amount. By using
some of Khinchin's ideas together with more modern results we are able to
provide a quantitative asymptotic theorem analogous to the classical one
mentioned above
Diophantine approximation and coloring
We demonstrate how connections between graph theory and Diophantine
approximation can be used in conjunction to give simple and accessible proofs
of seemingly difficult results in both subjects.Comment: 16 pages, pre-publication version of paper which will appear in
American Mathematical Monthl
Density of orbits of semigroups of endomorphisms acting on the Adeles
We investigate the question of whether or not the orbit of a point in A/Q,
under the natural action of a subset S of Q, is dense in A/Q. We prove that if
the set S is a multiplicative semigroup which contains at least two
multiplicatively independent elements, one of which is an integer, then the
orbit under S of any point with irrational real coordinate is dense.Comment: 13 page
Constructing bounded remainder sets and cut-and-project sets which are bounded distance to lattices
For any irrational cut-and-project setup, we demonstrate a natural infinite
family of windows which gives rise to separated nets that are each bounded
distance to a lattice. Our proof provides a new construction, using a
sufficient condition of Rauzy, of an infinite family of non-trivial bounded
remainder sets for any totally irrational toral rotation in any dimension.Comment: 11 pages, 1 figure, updated references, changed intro to give credit
to a result of Liardet which we were previously unaware o
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