Number Theory seminar

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