Jonas Jankauskas, Pure Math Department, University of Waterloo
“On the intersections of geometric and arithmetic progressions”
In my talk I present results from the joint paper [1]. We prove that the intersection G ∩ A of an infinite geometric progression
G = u,uq,uq2,uq3,..., q > 1,u > 0, and an infinite arithmetic progression
A = v, v + D, v + 3D, . . . ,
0 ≤ v < D
contains
at
most
3
elements
except
for
two
kinds
of
ratios
q.
The
first
exception
occurs
for
q
=
r1/d,
where
r
>
1
is
a
rational
number
and
d
∈
N.
Then
G
∩
A
can
be
of
any
cardinality
s
∈
N
or
infinite.
The
other
(possible)
exception
may
occur
for
q
=
β1/d,
where
β
>
1
is
a
real
cubic
algebraic
number
with
two
non
real
conjugates
of
moduli
distinct
from
β
and
d
∈
N.
In
this
(cubic)
case,
we
prove
that
the
intersection
|G
∩
A|
≤
6.
I will discuss how this problem can be transformed in to the language of fractional parts of powers of algebraic numbers {ξαn} and non-degenerate rational linear recurrent sequences. Time permitting, I will sketch the proof that is based on the geometry of complex zeros of trinomial equations and the results on the zero multiplicity in non-degenerate rational linear recurrent sequences.
References
[1] A. Dubickas and J. Jankauskas, On the intersection of infinite geometric and arithmetic progressions, Bull. of the Brazilian Math. Soc., 41 (4) (2010), 551–566.
MC 5136B