Constructing small subsets with a given packing index in Abelian groups
by
Nadia Lyaskovska
Ivan Franko Lviv National University

For a subset A of a group G, we consider the following two cardinal numbers

indP(A)= sup {|S|:S ⊂ G is such that {xA}x ∈ S is disjoint}

and

IndP(A)= sup {|S|:S ⊂ G is such that{xA}x ∈ S is almost disjoint},

called the packing indexes of A in G.

Those packing indexes have sharp versions carrying a bit more information about A:

ind+P(A)= sup {|S|+:S ⊂ G is such that {xA}x ∈ S is disjoint}

and

Ind+P(A)= sup {|S|+:S ⊂ G is such that{xA}x ∈ S is almost disjoint}.

It follows that ind_P(A)=sup{k:k < ind⁺_P(A)} and ind_P(A) ≤ Ind_P(A). The (sharp) packing indexes measure the geometric smallness of a subset of a group. Subsets with small packing index can be thought as large in a geometric sense.

Theorem 1 Let G be an infinite Abelian group and L ⊂ G be a subset with Ind_P(L)=1. For a cadinal k ∈ [2, |G|⁺] the following conditions are equivalent:

1) there is a subset A ⊂ G with ind_P⁺(A)=k;
2) there is a subset A ⊂ L with ind⁺_P(A)=Ind⁺_P(A)=k;
3) if |G/[G]₂| ≤ 2, then k ≠ 4 and if G=[G]₃, then k ≠ 3.

Here [G]_p={x ∈ G:x^p=1} stands for the subgroup of elements of order p.

Observe that the equality Ind_P(L)=1 is equivalent to |L∩(g+L)|=|G| for all g ∈ G, which means that L is rather large in a geometric sense. Still such a set L can be presented as union of small sets.

Theorem 2 Each infinite group G contains two subsets A, B ⊂ G such that ind_P⁺(A)=ind_P⁺(B)=|G|⁺ but Ind_P(A∪B)=1.

Date received: May 7, 2008