Abstract. Pilz’s conjecture states that for any finite set [Formula: see text] of positive integers and positive integer [Formula: see text] in the union of the sets [Formula: see text] (considered as a multiset) at least [Formula: see text] values appear an odd number of times. In this short note we consider a variant of this problem. Namely, we show that in the sumset [Formula: see text] (considered as a multiset) at least [Formula: see text] values appear an odd number of times.

We show that sets avoiding six-term arithmetic progressions in $\mathbb{Z}_6^n$ have size at most $5.709^n$. It is also pointed out that the “product construction” does not work in this setting; in particular we show that for the extremal sizes in small dimensions we have $r_6(\mathbb{Z}_6)=5$, $r_6(\mathbb{Z}_6^2)=25$, and $ 117\leq r_6(\mathbb{Z}_6^n)\leq 124$.

Abstract We study progression-free sets in the abelian groups $$G=({{\mathbb {Z}}}_m^n,+)$$ G = ( Z m n , + ) . Let $$r_k({{\mathbb {Z}}}_m^n)$$ r k ( Z m n ) denote the maximal size of a set $$S \subset {{\mathbb {Z}}}_m^n$$ S ⊂ Z m n that does not contain a proper arithmetic progression of length k. We give lower bound constructions, which e.g. include that $$r_3({{\mathbb {Z}}}_m^n) \ge C_m \frac{((m+2)/2)^n}{\sqrt{n}}$$ r 3 ( Z m n ) ≥ C m ( ( m + 2 ) / 2 ) n n , when m is even. When $$m=4$$ m = 4 this is of order at least $$3^n/\sqrt{n}\gg \vert G \vert ^{0.7924}$$ 3 n / n ≫ | G | 0.7924 . Moreover, if the progression-free set $$S\subset {{\mathbb {Z}}}_4^n$$ S ⊂ Z 4 n satisfies a technical condition, which dominates the problem at least in low dimension, then $$|S|\le 3^n$$ | S | ≤ 3 n 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: $$r_6({{\mathbb {Z}}}_{125}^n) \ge (85-o(1))^n$$ r 6 ( Z 125 n ) ≥ ( 85 - o ( 1 ) ) n . For $$r_3({{\mathbb {Z}}}_4^n)$$ r 3 ( Z 4 n ) we determine the exact values, when $$n \le 5$$ n ≤ 5 , e.g. $$r_3({{\mathbb {Z}}}_4^5)=124$$ r 3 ( Z 4 5 ) = 124 , and for $$r_4({{\mathbb {Z}}}_4^n)$$ r 4 ( Z 4 n ) we determine the exact values, when $$n \le 4$$ n ≤ 4 , e.g. $$r_4({{\mathbb {Z}}}_4^4)=128$$ r 4 ( Z 4 4 ) = 128 . With regard to affine caps, i.e. sets without 3 points on a line, the new methods asymptotically improve the known lower bounds, when $$m=4$$ m = 4 and $$m=5$$ m = 5 : in $${{\mathbb {Z}}}_4^n$$ Z 4 n from $$2.519^n$$ 2 . 519 n to $$(3-o(1))^n$$ ( 3 - o ( 1 ) ) n , and when $$m=5$$ m = 5 from $$2.942^n$$ 2 . 942 n to