Large subsets of mathbb {Z}_m^n without arithmetic progressions

Abstract

For integers m and n, we study the problem of finding good lower bounds for the size of progression-free sets in (mathbb {Z}_{m}^{n},+). Let r_{k}(mathbb {Z}_{m}^{n}) denote the maximal size of a subset of mathbb {Z}_{m}^{n} 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 r_{k}(mathbb {Z}_{m}^{n}):If kge 5 is odd and P^{-}(m)ge (k+2)/2, then rk(Zmn)≫m,k⌊k-1k+1m+1⌋nn⌊k-1k+1m⌋/2.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} r_k(\\mathbb {Z}_m^n) \\gg _{m,k} \\frac{\\bigl \\lfloor \\frac{k-1}{k+1}m +1\\bigr \\rfloor ^{n}}{n^{\\lfloor \\frac{k-1}{k+1}m \\rfloor /2}}. \\end{aligned}$$\\end{document}If kge 4 is even, P^{-}(m) ge k and m equiv -1 bmod k, then rk(Zmn)≫m,k⌊k-2km+2⌋nn⌊k-2km+1⌋/2.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} r_{k}(\\mathbb {Z}_{m}^{n}) \\gg _{m,k} \\frac{\\bigl \\lfloor \\frac{k-2}{k}m + 2\\bigr \\rfloor ^{n}}{n^{\\lfloor \\frac{k-2}{k}m + 1\\rfloor /2}}. \\end{aligned}$$\\end{document} Moreover, we give some further improved lower bounds on r_k(mathbb {Z}_p^n) for primes p le 31 and progression lengths 4 le k le 8.