Large Subsets of Z_m^n without Arithmetic Progressions

Christian Elsholtz*, Benjamin Klahn, Gabriel Friedrich Lipnik

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


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 languageEnglish
JournalDesigns, Codes, and Cryptography
Publication statusE-pub ahead of print - 15 Dec 2022


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