Almost arithmetic progressions in ℤ/pℤ

Mario Huicochea
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 24, 2018, Number 1, Pages 61–75
DOI: 10.7546/nntdm.2018.24.1.61-75
Full paper (PDF, 202 Kb)

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Authors and affiliations

Mario Huicochea
Facultad de Ciencias, UNAM-Juriquilla
Queretaro, Mexico

Abstract

For k ∈ ℕ ∪ {0} and r ∈ ℤ/pℤ \ {0}, we say that a subset X of ℤ/pℤ is a k-almost arithmetic progression with difference r if there is an arithmetic progression Y with difference r containing X such that |Y\X| ≤ k. Let X be a k-almost arithmetic progression with difference r such that k + 2 < |X| < p − 4k − 9. The main result of this paper is following: if there is t ∈ ℤ/pℤ \ {0} such that X is also a k-almost arithmetic progression with difference t, then t ∈ {±r}. Moreover, we will show that our result is sharp.

Keywords

• Arithmetic progressions
• Almost arithmetic progressions

2010 Mathematics Subject Classification

• Primary 11B13
• Secondary 11A07

References

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