807 research outputs found
On the degree of regularity of a certain quadratic Diophantine equation
We show that, for every positive integer r, there exists an integer b = b(r) such that the 4-variable quadratic
Diophantine equation (x1 − y1)(x2 − y2) = b is r-regular. Our proof uses Szemerédi’s theorem on arithmetic
progressions
Diophantine conditions and real or complex Brjuno functions
The continued fraction expansion of the real number x=a_0+x_0, a_0\in
{\ZZ}, is given by 0\leq x_n<1, x_{n}^{-1}=a_{n+1}+ x_{n+1}, a_{n+1}\in
{\NN}, for The Brjuno function is then
and the number
satisfies the Brjuno diophantine condition whenever is bounded.
Invariant circles under a complex rotation persist when the map is analytically
perturbed, if and only if the rotation number satisfies the Brjuno condition,
and the same holds for invariant circles in the semi-standard and standard maps
cases. In this lecture, we will review some properties of the Brjuno function,
and give some generalisations related to familiar diophantine conditions. The
Brjuno function is highly singular and takes value on a dense set
including rationals. We present a regularisation leading to a complex function
holomorphic in the upper half plane. Its imaginary part tends to the Brjuno
function on the real axis, the real part remaining bounded, and we also
indicate its transformation under the modular group.Comment: latex jura.tex, 6 files, 19 pages Proceedings on `Noise, Oscillators
and Algebraic Randomness' La Chapelle des Bois, France 1999-04-05 1999-04-10
April 5-10, 1999 [SPhT-T99/116
Persistence of Diophantine flows for quadratic nearly-integrable Hamiltonians under slowly decaying aperiodic time dependence
The aim of this paper is to prove a Kolmogorov-type result for a
nearly-integrable Hamiltonian, quadratic in the actions, with an aperiodic time
dependence. The existence of a torus with a prefixed Diophantine frequency is
shown in the forced system, provided that the perturbation is real-analytic and
(exponentially) decaying with time. The advantage consists of the possibility
to choose an arbitrarily small decaying coefficient, consistently with the
perturbation size.Comment: Several corrections in the proof with respect to the previous
version. Main statement unchange
Best possible rates of distribution of dense lattice orbits in homogeneous spaces
The present paper establishes upper and lower bounds on the speed of
approximation in a wide range of natural Diophantine approximation problems.
The upper and lower bounds coincide in many cases, giving rise to optimal
results in Diophantine approximation which were inaccessible previously. Our
approach proceeds by establishing, more generally, upper and lower bounds for
the rate of distribution of dense orbits of a lattice subgroup in a
connected Lie (or algebraic) group , acting on suitable homogeneous spaces
. The upper bound is derived using a quantitative duality principle for
homogeneous spaces, reducing it to a rate of convergence in the mean ergodic
theorem for a family of averaging operators supported on and acting on
. In particular, the quality of the upper bound on the rate of
distribution we obtain is determined explicitly by the spectrum of in the
automorphic representation on . We show that the rate
is best possible when the representation in question is tempered, and show that
the latter condition holds in a wide range of examples
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