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Answers are given to two questions concerning the existence of some sparse subsets of H = f0; 1;:::;H, 1g# # N, where H is a hyper#nite integer. In x1, we answer a question of Kanovei by showing that for a given cut U in H, there exists a countably determined set X #Hwhich contains exactly one element in each U-monad, if and only if U = a # N for some a 2 H r f0g. In x2, we deal with a question of Keisler and Leth in #6#. We show that there is a cut V #Hsuch that for any cut U , #i# there exists a U-discrete set X #Hwith X +X = H #mod H# provided U $ V , #ii# there does not exist any U-discrete set X #Hwith X+X = H #mod H# provided U # V . We obtain some partial results for the case U = V . 0 Notation and De#nition We always work within a #xed ! 1 -saturated nonstandard universe # V in the sense of #1#. The reader is assumed to be familiar with the basic de#nitions and facts about nonstandard universe and nonstandard analysis. Those de#nitions and facts could be found in x4.4..

Year: 2001

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