TY - JOUR

T1 - Large Subsets of Z_m^n without Arithmetic Progressions

AU - Elsholtz, Christian

AU - Klahn, Benjamin

AU - Lipnik, Gabriel Friedrich

PY - 2022/12/15

Y1 - 2022/12/15

N2 - 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.

AB - 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.

U2 - 10.1007/s10623-022-01145-w

DO - 10.1007/s10623-022-01145-w

M3 - Article

JO - Designs, Codes and Cryptography

JF - Designs, Codes and Cryptography

SN - 0925-1022

ER -