David Ralston
Continued fractions and heavy sequences.
Thu, 11/12/2009 - 16:01 — SomNambulAuthors: David Ralston, Michael Boshernitzan
Subjects: Number Theory
AbstractWe initiate the study of the sets $H(c)$, $0<c<1$, of real $x$ for which the
sequence $(kx)_{k\geq1}$ (viewed mod 1) consistently hits the interval $[0,c)$
at least as often as expected (i. e., with frequency $\geq c$). More formally,
\[ H(c)=\{\alpha\in \mathbf R\mid {\rm card}(\{1\leq k\leq n\mid <
k\alpha><c\})\geq cn, {for all}n\geq1\}. \] where $<x>=x-[x]$ stands for the
fractional part of $x\in \mathbb R$.On the Frequency of Balanced Times in Cylinder Flows.
Tue, 09/01/2009 - 12:47 — SomNambulAuthors: David Ralston, Jon Chaika
Subjects: Dynamical Systems
AbstractGiven an irrational alpha and an x in the unit interval, the set of balanced
times, for which the same number of (k*alpha+x) (modulo one) are less than or
equal to one half as are larger than one half, is in general infinite, but
sparse in terms of density. We investigate the sparseness of this sequence in
terms of summation over reciprocals. Our results are that for the generic pair
(alpha,x), the resulting sum diverges, but there are certain exceptional alpha
for which the associated sums converge for every x.Controlled Divergence of Discrepancy Sums.
Tue, 08/25/2009 - 22:41 — SomNambulAuthors: David Ralston
Subjects: Number Theory
AbstractAnswering an informal question of K. Park, we show that by fixing some
irrational alpha to have a particular standard continued fraction expansion, we
may force the associated discrepancy sequences for all x in [0,1), which track
the difference between the number of values in the orbit of x under rotation by
alpha (modulo one) less than one half versus the number larger than one half,
to have maximal values which grow at a prescribed rate.
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