Event Detail

Borel (1919) defined a set of real numbers A to have strong measure zero if for every sequence of positive numbers (epsilon_i: i in omega) there is an open cover of A, (U_i: i in omega), such that for each i, the diameter of U_i is less than epsilon_i. Besicovitch (1956) showed that A has strong measure zero if and only if A has strong dimension zero, which means that for every gauge function f, A is null for its associated measure H^f. We say that a subset of A of R^n has strong dimension f if and only if H^f(A)>0 and for every gauge function g of higher order H^g(A)=0. Here, g has higher order than f when lim_{t to 0^+} g(t)/f(t)=0.

Borel conjectured that a set of strong measure zero must be countable. This conjecture naturally extends to the assertion that a set has strong dimension f if and only if it is sigma-finite for H^f. Sierpinski (1928) used the continuum hypothesis to give a counterexample to Borel’s conjecture and Besicovitch (1963) did the same for its generalization. Laver (1976) showed that Borel’s conjecture is relatively consistent with ZFC, the conventional axioms of set theory including the axiom of choice. We will discuss the proof that its generalization to strong dimension is also relatively consistent with ZFC.