Caps and progression-free sets in

We study progression-free sets in the abelian groups $$ . Let $$ denote the maximal size of a set $$ that does not contain a proper arithmetic progression of length k. We give lower bound constructions, which e.g. include that $$ , when m is even. When $$ this is of order at least $$ . Moreover, if the progression-free set $$ satisfies a technical condition, which dominates the problem at least in low dimension, then $$ holds. We present a number of new methods which cover lower bounds for several infinite families of parameters m, k, n, which includes for example: $$ . For $$ we determine the exact values, when $$ , e.g.  $$ , and for $$ we determine the exact values, when $$ , e.g.  $$ . With regard to affine caps, i.e. sets without 3 points on a line, the new methods asymptotically improve the known lower bounds, when $$ and $$ : in $$ from $$ to $$ , and when $$ from $$ to $$ . This last improvement modulo 5 appears to be the first asymptotic improvement of any cap in AG(n, m), when $$ over a tensor lifting from dimension 6 (see Edel, in Des Codes Crytogr 31:5-14, 2004).