On Classification of Sequences Containing Arbitrarily Long Arithmetic Progressions

Celik S. C., Eyidogan S., Goral H., Sertbas D. C.

INTERNATIONAL JOURNAL OF NUMBER THEORY, cilt.19, sa.8, ss.1917-1952, 2023 (SCI-Expanded)

  • Yayın Türü: Makale / Tam Makale
  • Cilt numarası: 19 Sayı: 8
  • Basım Tarihi: 2023
  • Doi Numarası: 10.1142/s1793042123500926
  • Dergi Adı: INTERNATIONAL JOURNAL OF NUMBER THEORY
  • Derginin Tarandığı İndeksler: Science Citation Index Expanded (SCI-EXPANDED), Scopus, Academic Search Premier, MathSciNet, zbMATH
  • Sayfa Sayıları: ss.1917-1952
  • Çukurova Üniversitesi Adresli: Hayır

Özet

In this paper, we study the classification of sequences containing arbitrarily long arithmetic progressions. First, we deal with the question how the polynomial map n^s can be extended so that it contains arbitrarily long arithmetic progressions. Under some growth conditions, we construct sequences which contain arbitrarily long arithmetic progressions. Also, we give a uniform and explicit arithmetic progression rank bound for a large class of sequences. Consequently, a dichotomy result is deduced on the finiteness of the arithmetic progression rank of certain sequences. Therefore, in this paper, we see a way to determine the finiteness of the arithmetic progression rank of various sequences satisfying some growth conditions.