Discordant sets and ergodic Ramsey theory

The paper proves that for any countable abelian group G, any Følner sequence Φ, and any c in (0,1), there exists a Straus (anti-recurrent) subset A with d_Φ(A)=c, first for G=Z (Theorem 36) and then for arbitrary countable abelian G (Theorem 37). The proof builds a compact group X with a dense homomorphism ϕ:G→X, uses unique ergodicity of the translation action to identify densities via Haar measure, then carves out A so that every translate fails recurrence by avoiding small neighborhoods around 0 (see the construction and density computation , and the unique ergodicity/equidistribution input ; also the summability/cofinite-selection lemma used in the union step ). The candidate solution proves the same result by a different route: it embeds G into its Bohr compactification inside a product of circles, proves equidistribution of Φ via character-averages, chooses a Borel set B of Haar measure c avoiding a prescribed small union of translates, and sets A=ι^{-1}(B). Anti-recurrence is then witnessed in the same compact rotation system by selecting, for each translate, a small neighborhood Y with Y∩hY=∅ for all h in that translate. The two proofs reach the same conclusion; they differ mainly in the compact model and the way the ‘forbidden’ set is assembled.