Large entire cross-sections of second category sets ...

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Title Large entire cross-sections of second category sets in Rn+1
Author(s) Maxim R. Burke
Journal Topology and its Applications
Date 2007
Volume 154
Issue 1
Start page 215
End page 240
Abstract By the Kuratowski-Ulam theorem, if A subset of Rn+1 = R-n x R is a Borel set which has second category intersection with every ball (i.e., is "everywhere second category"), then there is a y is an element of R such that the section A boolean AND (R-n x {y}) is everywhere second category in R-n x {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 subset of Rn+1 such that for any polynomial p in n variables, A boolean AND graph(p) is finite. It is a classical result that under the Continuum Hypothesis, there is an everywhere second category set L in Rn+1 which has only countably many points in any first category set. In particular, L boolean AND 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 Rn+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 n graph(f) is everywhere second category. For any collection of less than 2(N)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 epsilon:R-n -> R, we may choose f so that for all multiindices alpha of order at most k and for all X is an element of R-n, vertical bar D-alpha g(x) - D(alpha)g(x)vertical bar <epsilon(x). The method builds on fundamental work of K. Ciesielski and S. Shelah which provides, for everywhere second category sets in 2(omega) x 2(omega), large sections which are the graphs of homeomorphisms of 2(omega). K. Ciesielski and T. Natkaniec adapted the Ciesielski-Shelah result for subsets of R x R and proved the existence in this setting of large sections which are increasing homeormorphisms 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. (c) 2006 Elsevier B.V. All rights reserved.

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