We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.
To send content items to your account,
please confirm that you agree to abide by our usage policies.
If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account.
Find out more about sending content to .
To send content items to your Kindle, first ensure no-reply@cambridge.org
is added to your Approved Personal Document E-mail List under your Personal Document Settings
on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part
of your Kindle email address below.
Find out more about sending to your Kindle.
Note you can select to send to either the @free.kindle.com or @kindle.com variations.
‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi.
‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.
We carry Sprindžuk’s classification of the complex numbers to the field
$\mathbb{Q}_{p}$
of
$p$
-adic numbers. We establish several estimates for the
$p$
-adic distance between
$p$
-adic roots of integer polynomials, which we apply to show that almost all
$p$
-adic numbers, with respect to the Haar measure, are
$p$
-adic
$\tilde{S}$
-numbers of order 1.
In the field
$\mathbb{K}$
of formal power series over a finite field
$K$
, we consider some lacunary power series with algebraic coefficients in a finite extension of
$K(x)$
. We show that the values of these series at nonzero algebraic arguments in
$\mathbb{K}$
are
$U$
-numbers.
We exhibit the first explicit examples of Salem sets in ℚp of every dimension 0 < α < 1 by showing that certain sets of well-approximable p-adic numbers are Salem sets. We construct measures supported on these sets that satisfy essentially optimal Fourier decay and upper regularity conditions, and we observe that these conditions imply that the measures satisfy strong Fourier restriction inequalities. We also partially generalize our results to higher dimensions. Our results extend theorems of Kaufman, Papadimitropoulos, and Hambrook from the real to the p-adic setting.
We show that two distinct singular moduli
$j(\unicode[STIX]{x1D70F}),j(\unicode[STIX]{x1D70F}^{\prime })$
, such that for some positive integers
$m$
and
$n$
the numbers
$1,j(\unicode[STIX]{x1D70F})^{m}$
and
$j(\unicode[STIX]{x1D70F}^{\prime })^{n}$
are linearly dependent over
$\mathbb{Q}$
, generate the same number field of degree at most two. This completes a result of Riffaut [‘Equations with powers of singular moduli’, Int. J. Number Theory, to appear], who proved the above theorem except for two explicit pairs of exceptions consisting of numbers of degree three. The purpose of this article is to treat these two remaining cases.
We give transcendence measures for
$p$
-adic numbers
$\unicode[STIX]{x1D709}$
, having good rational (respectively, integer) approximations, that force them to be either
$p$
-adic
$S$
-numbers or
$p$
-adic
$T$
-numbers.
Let
$\Vert \cdot \Vert$
denote the distance to the nearest integer and, for a prime number
$p$
, let
$|\cdot |_{p}$
denote the
$p$
-adic absolute value. Over a decade ago, de Mathan and Teulié [Problèmes diophantiens simultanés, Monatsh. Math. 143 (2004), 229–245] asked whether
$\inf _{q\geqslant 1}$
$q\cdot \Vert q{\it\alpha}\Vert \cdot |q|_{p}=0$
holds for every badly approximable real number
${\it\alpha}$
and every prime number
$p$
. Among other results, we establish that, if the complexity of the sequence of partial quotients of a real number
${\it\alpha}$
grows too rapidly or too slowly, then their conjecture is true for the pair
$({\it\alpha},p)$
with
$p$
an arbitrary prime.
The aim of this work is to adapt a construction of the so-called
$U_{m}$
-numbers (
$m\gt 1$
), which are extended Liouville numbers with respect to algebraic numbers of degree
$m$
but not with respect to algebraic numbers of degree less than
$m$
, to the
$p$
-adic frame.
Let p be a prime number. For a positive integer n and a p-adic number ξ, let λn(ξ) denote the supremum of the real numbers λ such that there are arbitrarily large positive integers q such that ‖qξ‖p,‖qξ2‖p,…,‖qξn‖p are all less than q−λ−1. Here, ‖x‖p denotes the infimum of |x−n|p as n runs through the integers. We study the set of values taken by the function λn.
The chain complexes underlying Floer homology theories typically carry a real-valued filtration, allowing one to associate to each Floer homology class a spectral number defined as the infimum of the filtration levels of chains representing that class. These spectral numbers have been studied extensively in the case of Hamiltonian Floer homology by Oh, Schwarz and others. We prove that the spectral number associated to any nonzero Floer homology class is always finite, and that the infimum in the definition of the spectral number is always attained. In the Hamiltonian case, this implies that what is known as the ‘nondegenerate spectrality’ axiom holds on all closed symplectic manifolds. Our proofs are entirely algebraic and apply to any Floer-type theory (including Novikov homology) satisfying certain standard formal properties. The key ingredient is a theorem about the existence of best approximations of arbitrary elements of finitely generated free modules over Novikov rings by elements of prescribed submodules with respect to a certain family of non-Archimedean metrics.
We generalise the approximation theory described in Mahier's paper “Perfect Systems” to linked simultaneous approximations and prove the existence of nonsingular approximation and of transfer matrices by generalising Coates' normality zig-zag theorem. The theory sketched here may have application to constructions important in the theory of diophantine approximation.
The study of the S-unit equation for algebraic numbers rests very heavily on Schmidt's Subspace Theorem. Here we prove an effective subspace theorem for the differential function field case, which should be valuable in the proof of results concerning the S-unit equation for function fields. Theorem 1 states that either has a given upper bound where are linearly independent linear forms in the polynomials with coefficients that are formal power series solutions about x = 0 of non-zero differential equations and where Orda denotes the order of vanishing about a regular (finite) point of functions ƒk, i: (k = 1, n; i = 1, n) or lies inside one of a finite number of proper subspaces of (K(x))n. The proof of the theorem is based on the wroskian methods and graded sub-rings of Picard-Vessiot extensions developed by D. V. Chudnovsky and G. V. Chudnovsky in their function field analogues of the Roth and Schmidt theorems. A brief discussion concerning the possibility of a subspace theorem for a product of valuations including the infinite one is also included.
Recommend this
Email your librarian or administrator to recommend adding this to your organisation's collection.