12,477 research outputs found
The distribution of spacings between the fractional parts of
We study the distribution of normalized spacings between the fractional parts
of an^2, n=1,2,.... We conjecture that if a is "badly approximable" by
rationals, then the sequence of fractional parts has Poisson spacings, and give
a number of results towards this conjecture. We also present an example of a
Diophantine number a for which the higher correlation functions of the sequence
blow up.Comment: Substantial revisions in the exposition. Accepted for publication in
Inventiones mathematica
Ternary expansions of powers of 2
Paul Erdos asked how frequently the ternary expansion of 2^n omits the digit
2. He conjectured this happens only for finitely many values of n. We
generalize this question to consider iterates of two discrete dynamical
systems. The first is over the real numbers, and considers the integer part of
lambda 2^n for a real input lambda. The second is over the 3-adic integers, and
considers the sequence lambda 2^n for a 3-adic integer input lambda.
We show that the number of input values that have infinitely many iterates
omitting the digit 2 in their ternary expansion is small in a suitable sense.
For each nonzero input we give an asymptotic upper bound on the number of the
first k iterates that omit the digit 2, as k goes to infinity. We also study
auxiliary problems concerning the Hausdorff dimension of intersections of
multiplicative translates of 3-adic Cantor sets.Comment: 28 pages latex; v4 major revision, much more detail to proofs, added
material on intersections of Cantor set
Scaling and non-Abelian signature in fractional quantum Hall quasiparticle tunneling amplitude
We study the scaling behavior in the tunneling amplitude when quasiparticles
tunnel along a straight path between the two edges of a fractional quantum Hall
annulus. Such scaling behavior originates from the propagation and tunneling of
charged quasielectrons and quasiholes in an effective field analysis. In the
limit when the annulus deforms continuously into a quasi-one-dimensional ring,
we conjecture the exact functional form of the tunneling amplitude for several
cases, which reproduces the numerical results in finite systems exactly. The
results for Abelian quasiparticle tunneling is consistent with the scaling
anaysis; this allows for the extraction of the conformal dimensions of the
quasiparticles. We analyze the scaling behavior of both Abelian and non-Abelian
quasiparticles in the Read-Rezayi Z_k-parafermion states. Interestingly, the
non-Abelian quasiparticle tunneling amplitudes exhibit nontrivial k-dependent
corrections to the scaling exponent.Comment: 16 pages, 4 figure
Modular Groups, Visibility Diagram and Quantum Hall Effect
We consider the action of the modular group on the set of
positive rational fractions. From this, we derive a model for a classification
of fractional (as well as integer) Hall states which can be visualized on two
``visibility" diagrams, the first one being associated with even denominator
fractions whereas the second one is linked to odd denominator fractions. We use
this model to predict, among some interesting physical quantities, the relative
ratios of the width of the different transversal resistivity plateaus. A
numerical simulation of the tranversal resistivity plot based on this last
prediction fits well with the present experimental data.Comment: 17 pages, plain TeX, 4 eps figures included (macro epsf.tex), 1
figure available from reques
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