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 email@example.com
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 give an integrability criterion on a real-valued non-increasing function
guaranteeing that for almost all (or almost no) pairs
is a real
, the system
is solvable in
for all sufficiently large
. The proof consists of a reduction to a shrinking target problem on the space of grids in
. We also comment on the homogeneous counterpart to this problem, whose
case was recently solved, but whose general case remains open.
be a non-increasing function. A real number
is said to be
-Dirichlet improvable if it admits an improvement to Dirichlet’s theorem in the following sense: the system
has a non-trivial integer solution for all large enough
. Denote the collection of such points by
. In this paper we prove that the Hausdorff measure of the complement
(the set of
-Dirichlet non-improvable numbers) obeys a zero-infinity law for a large class of dimension functions. Together with the Lebesgue measure-theoretic results established by Kleinbock and Wadleigh [A zero-one law for improvements to Dirichlet’s theorem. Proc. Amer. Math. Soc.146 (2018), 1833–1844], our results contribute to building a complete metric theory for the set of Dirichlet non-improvable numbers.
Email your librarian or administrator to recommend adding this to your organisation's collection.