Higher Randomness and Forcing with Closed Sets.

Kechris showed in Kechris (Trans. Am. Math. Soc. 202, 259-297, 1975) that there exists a largest ${\Pi ^{1}_{1}}$ set of measure 0. An explicit construction of this largest ${\Pi ^{1}_{1}}$ nullset has later been given in Hjorth and Nies (J. Lond. Math. Soc. 75(2), 495-508, 2007). Due to its universal nature, it was conjectured by many that this nullset has a high Borel rank (the question is explicitely mentioned in Chong and Yu (J. Symb. Log. 80(04), 1131-1148, 2015) and Yu (Fundam. Math. 215, 219-231, 2011)). In this paper, we refute this conjecture and show that this nullset is merely ${\Sigma }^{0}_{3}$ . Together with a result of Liang Yu, our result also implies that the exact Borel complexity of this set is ${\Sigma }^{0}_{3}$ . To do this proof, we develop the machinery of effective randomness and effective Solovay genericity, investigating the connections between those notions and effective domination properties. [ABSTRACT FROM AUTHOR]