Zoom, Online seminar
(線上演講 Zoom)
Fibers on Well Approximable Set of Systems of Affine Forms
Bo Wang (South China University of Technology)
Abstract
Let
and
. Denote by
the set of all
such that
for infinitely many
, where ||• ||denotes supremum norm and
represents supremum norm distance to
. Let
and
be the sets of all decreasing function
satisfying
diverges and converges respectively. In this paper, we prove that the union of all the fibers of
for
is equal to the complement of badly approximable set. We also give some metrical results of the fibers of
for given
and
. Our result gives partial answer to Ramírez's questions (2018), he asked whether the fiber of
for any given
is non empty and whether the fiber of
for some
is of full measure. Then we show that
is equal to the complement of the set of badly approximable systems. Moreover, let
, we study the topological structure of the set of all
with
for infinitely many
.