Sparse, thin and P-small subsets of infinite groups
by
Eugen Lutsenko
Department of Cybernetics, Kyiv University, UKRAINE
Coauthors: I.V. Protasov

A subset A of an infinite group G with the identity e is said to be

• thin if gA∩A is finite for every g ∈ G, g ≠ e;

• k-thin for k ∈ N if |gA∩A| ≤ k for each g ∈ G, g ≠ e;

• almost thin if D(A)={g ∈ G:gA∩A is infinite} is finite;

• sparse if, for every infinite subset X ⊆ G, there exists a non-empty finite subset F ⊂ X such that ∩_{g ∈ F}gA is finite;

• k-sparse for k ∈ N if, for every infinite subset X of G, there exists a non-empty finite subset F of X such that |F| ≤ k and ∩_{g ∈ F}gA is finite;

• P-small if there exists an injective sequence (g_n)_{n ∈ w} in G such that the subsets (g_nA)_{n ∈ w} are pairwise disjoint;

• almost P-small if there exists an injective sequence (g_n)_{n ∈ w} in G such that g_iA∩g_jA is finite for all distinct i, j ∈ w.

Theorem 1 For every group G and every k ∈ N, there exist (k+1)-sparse but not k-sparse subset of G.

Theorem 2 For every group G, there exists a sparse subset of G which is not k-sparse for every k ∈ w.

Theorem 3 For every group G, there exists a thin subset of G which is not k-thin for every k ∈ w.

Theorem 4 Every group G can be generated by some 2-thin subset.

Theorem 5 Every almost thin subset A of a group G can be partitioned in 3^|D(A)|-1 thin subsets. If G has no elements of odd order, then A can be partitioned in 2^|D(A)|-1 thin subsets.

Theorem 6 Every almost thin subset of a group G is 2-sparse. Every 2-sparse subset of a group G is almost P-small. Every almost P-small subset can be partitioned in two P-small subsets.

Theorem 7 For every group G, there exists a 2-sparse subset which cannot be partitioned in finitely many thin subsets.

Theorem 8 For every group G, there exists a 2-thin subset which is not P-small.

Theorem 9 For every group G, there exists a P-small subset which is not sparse.

Theorem 10 If a subset A of a group G is either sparse or almost P-small, then m(A)=0 for each left invariant Banach measure m on G.

Date received: March 26, 2008