## Contact Info

Pure MathematicsUniversity of Waterloo

200 University Avenue West

Waterloo, Ontario, Canada

N2L 3G1

Departmental office: MC 5304

Phone: 519 888 4567 x33484

Fax: 519 725 0160

Email: puremath@uwaterloo.ca

Visit our COVID-19 information website to learn how Warriors protect Warriors.

Please note: The University of Waterloo is closed for all events until further notice.

Thursday, February 27, 2014 — 1:30 PM EST

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

Location

MC - Mathematics & Computer Building

5136B

200 University Avenue West

Waterloo, ON N2L 3G1

Canada

200 University Avenue West

Waterloo, ON N2L 3G1

Canada

University of Waterloo

200 University Avenue West

Waterloo, Ontario, Canada

N2L 3G1

Departmental office: MC 5304

Phone: 519 888 4567 x33484

Fax: 519 725 0160

Email: puremath@uwaterloo.ca

University of Waterloo

University of Waterloo

43.471468

-80.544205

200 University Avenue West

Waterloo,
ON,
Canada
N2L 3G1

The University of Waterloo acknowledges that much of our work takes place on the traditional territory of the Neutral, Anishinaabeg and Haudenosaunee peoples. Our main campus is situated on the Haldimand Tract, the land granted to the Six Nations that includes six miles on each side of the Grand River. Our active work toward reconciliation takes place across our campuses through research, learning, teaching, and community building, and is centralized within our Indigenous Initiatives Office.