Assuming V =L, we construct a plane set E of Hausdorff dimension 1 whose every orthogonal projection onto straight lines through the origin has Hausdorff dimension 0. This is a counterexample to J. M. Marstrand's seminal projection theorem [30]. While counterexamples had already been constructed decades ago, initially by R. O. Davies [8], the novelty of our result lies in the fact that E is co-analytic. Following Marstrand's original proof [30] (and R. Kaufman's newer, and now standard, approach [18] based on capacities), a counterexample to the projection theorem cannot be analytic, hence our counterexample is optimal. We then extend the result in a strong way: we show that for each ∈ (0, 1) there exists a co-analytic set E of dimension 1 + , each of whose orthogonal projections onto straight lines through the origin has Hausdorff dimension . The constructions of E and E are by induction on the countable ordinals, applying a theorem by Z. Vidnyánszky [42].

We extend results found by Greenberg, Turetsky, and Westrick in [7] and investigate effective properties of bases of uncountable free abelian groups. Assuming V = L, we show that if κ is a regular uncountable cardinal and X is a ∆11(Lκ) subset of κ, then there is a κ-computable free abelian group whose bases cannot be effectively computed by X. Unlike in [7], we give a direct construction.

We extend results found by Greenberg, Turetsky, and Westrick in [7] and investigate effective properties of bases of uncountable free abelian groups. Assuming V = L, we show that if κ is a regular uncountable cardinal and X is a ∆11(Lκ) subset of κ, then there is a κ-computable free abelian group whose bases cannot be effectively computed by X. Unlike in [7], we give a direct construction.