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)**

## Details

### Authors and affiliations

Mario Huicochea

*Facultad de Ciencias, UNAM-Juriquilla
Queretaro, Mexico
*

### Abstract

For *k* ∈ ℕ ∪ {0} and *r* ∈ ℤ/*p*ℤ \ {0}, we say that a subset *X* of ℤ/*p*ℤ 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* − 4*k* − 9. The main result of this paper is following: if there is *t* ∈ ℤ/*p*ℤ \ {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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*p*, Acta Arithmetica, 92, 251–262. - Hamidoune, Y., Serra, O., & Zemor, G. (2006) On the critical pair theory in Z/pZ, Acta Arithmetica, 121, 99–115.
- Huicochea, M., & Montejano, A. (2015) The structure of rainbow-free colorings for linear equations on three variables in Z
*p*, Integers: Electronic Journal of Combinatorial Number Theory, 15A, A8. - Neukirch, J. (2002) Algebraic Number Theory (Grundlehren der mathematischen Wissenschaften, Vol. 322, Spinger-Verlag.
- Vosper, G. (1956) The critical pairs of subsets of a group of prime order, J. London Math. Soc., 31, 200–205.

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## Cite this paper

Huicochea, M. (2018). Almost arithmetic progressions in ℤ/pℤ . *Notes on Number Theory and Discrete Mathematics*, 24(1), 61-75, DOI: 10.7546/nntdm.2018.24.1.61-75.