Large entire cross-sections of second category sets in [formula omitted

Abstract

By the Kuratowski–Ulam theorem, if A ⊆ R n + 1 = R n × R is a Borel set which has second category intersection with every ball (i.e., is “everywhere second category”), then there is a y ∈ R such that the section A ∩ ( R n × { y } ) is everywhere second category in R n × { y } . If A is not Borel, then there may not exist a large cross-section through A, even if the section does not have to be flat. For example, a variation on a result of T. Bartoszynski and L. Halbeisen shows that there is an everywhere second category set A ⊆ R n + 1 such that for any polynomial p in n variables, A ∩ graph ( p ) is finite. It is a classical result that under the Continuum Hypothesis, there is an everywhere second category set L in R n + 1 which has only countably many points in any first category set. In particular, L ∩ graph ( f ) is countable for any continuous function f : R n → R . We prove that it is relatively consistent with ZFC that for any everywhere second category set A in R n + 1 , there is a function f : R n → R which is the restriction to R n of an entire function on C n and is such that, relative to graph ( f ) , the set A ∩ graph ( f ) is everywhere second category. For any collection of less than 2 ℵ 0 sets A, the function f can be chosen to work for all sets A in the collection simultaneously. Moreover, given a nonnegative integer k, a function g : R n → R of class C k and a positive continuous function ε : R n → R , we may choose f so that for all multiindices α of order at most k and for all x ∈ R n , | D α f ( x ) − D α g ( x ) | < ε ( x ) . The method builds on fundamental work of K. Ciesielski and S. Shelah which provides, for everywhere second category sets in 2 ω × 2 ω , large sections which are the graphs of homeomorphisms of 2 ω . K. Ciesielski and T. Natkaniec adapted the Ciesielski–Shelah result for subsets of R × R and proved the existence in this setting of large sections which are increasing homeomorphisms of R . The technique used in this paper extends to functions of several variables an approach developed for functions of a single variable in previous related work of the author.