Abstract
For integers m and n, we study the problem of finding good lower bounds for the size of progression-free sets in (Zmn,+). Let rk(Zmn) denote the maximal size of a subset of Zmn without arithmetic progressions of length k and let P-(m) denote the least prime factor of m. We construct explicit progression-free sets and obtain the following improved lower bounds for rk(Zmn):If k≥ 5 is odd and P-(m) ≥ (k+ 2) / 2 , then (Formula presented.)If k≥ 4 is even, P-(m) ≥ k and m≡-1modk, then (Formula presented.) Moreover, we give some further improved lower bounds on rk(Zpn) for primes p≤ 31 and progression lengths 4 ≤ k≤ 8.
Original language | English |
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Pages (from-to) | 1443-1452 |
Number of pages | 10 |
Journal | Designs, Codes, and Cryptography |
Volume | 91 |
Issue number | 4 |
Early online date | 15 Dec 2022 |
DOIs | |
Publication status | Published - Apr 2023 |
Keywords
- Arithmetic progressions
- Behrend-type construction
- Progression-free sets
ASJC Scopus subject areas
- Theoretical Computer Science
- Computer Science Applications
- Discrete Mathematics and Combinatorics
- Applied Mathematics