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
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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

  1. Hamidoune, Y., & Rødseth, Ø. (2000) An inverse theorem mod p, Acta Arithmetica, 92, 251–262.
  2. Hamidoune, Y., Serra, O., & Zemor, G. (2006) On the critical pair theory in Z/pZ, Acta Arithmetica, 121, 99–115.
  3. Huicochea, M., & Montejano, A. (2015) The structure of rainbow-free colorings for linear equations on three variables in Zp, Integers: Electronic Journal of Combinatorial Number Theory, 15A, A8.
  4. Neukirch, J. (2002) Algebraic Number Theory (Grundlehren der mathematischen Wissenschaften, Vol. 322, Spinger-Verlag.
  5. 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

APA

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.

Chicago

Huicochea, Mario. “Almost Arithmetic Progressions in ℤ/pℤ.” Notes on Number Theory and Discrete Mathematics 24, no. 1 (2018): 61-75, doi: 10.7546/nntdm.2018.24.1.61-75.

MLA

Huicochea, Mario. “Almost Arithmetic Progressions in ℤ/pℤ.” Notes on Number Theory and Discrete Mathematics 24.1 (2018): 61-75. Print, doi: 10.7546/nntdm.2018.24.1.61-75.

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