Strongly Meager Sets and Their Uniformly Continuous Images

We prove the following theorems: (1) Suppose that f : 2ω → 2ω is a continuous function and X is a Sierpiński set. Then (A) for any strongly measure zero set Y , the image f [X + Y ] is an s0-set, (B) f [X] is a perfectly meager set in the transitive sense. (2) Every strongly meager set is completely Ramsey null. This paper is a continuation of earlier works by the authors and by M. Scheepers (see [N], [NSW], [S]) in which properties (mainly, the algebraic sum) of certain singular subsets of the real line R and of the Cantor set 2 were investigated. Throughout the paper, by a set of real numbers we mean a subset of 2 and by “+” we denote the standard modulo 2 coordinatewise addition in 2. Let us also assume that a “measure zero” (or “negligible”) set always denotes a Lebesgue measure zero set. We apply the following definition of sets of real numbers. Definition 1. An uncountable set X is said to be a Luzin (respectively, Sierpiński) set iff for each meager (respectively, measure zero) set Y , X∩Y is at most countable. We say that a set X is of strong measure zero (respectively, strongly meager) iff for each meager (respectively, measure zero) set Y , X + Y 6= 2. Remark 1. It is well known (see [M] for example) that every Luzin set is strongly measure zero. Quite recently J. Pawlikowski proved that each Sierpiński set must be strongly meager as well (see [P]). Let us recall that a set X is called an s0-set (or Marczewski set) iff for each perfect set P one can find a perfect set Q⊆P that is disjoint from X . M. Scheepers showed in [S] that for a Sierpiński set X and a strong measure zero set Y , X + Y is an s0-set. Later, in [NSW] it was proven that this also holds when X is strongly meager. We have the following functional version of the M. Scheepers’ result. Theorem 1. Let X be a Sierpiński set and let Y be a strong measure zero set. Assume also that f : 2 → 2 is a continuous function. Then the image f [X + Y ] is an s0-set. Received by the editors July 16, 1998 and, in revised form, September 9, 1998 and March 10, 1999. 2000 Mathematics Subject Classification. Primary 03E15, 03E20, 28E15.