A set tiles the integers if and only if the integers can be written as a disjoint union of translates of that set. We consider the problem of finding necessary and sufficient conditions for a finite set to tile the integers. For sets of prime power Ž . size, it was solved by D. Newman 1977, J. Number Theory 9, 107᎐111 . We solve it for sets of size having at most two prime factors. The conditions are always sufficient, but it is unknown whether they are necessary for all finite sets.

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ABSTRACT. We study the family of tent maps-continuous, unimodal, piecewise linear maps of the interval with slopes ±s, \/2 < s < 2. We show that tent maps have the shadowing property (every pseudo-orbit can be approximated by an actual orbit) for almost all parameters s, although they fail to have the shadowing property for an uncountable, dense set of parameters.We also show that for any tent map, every pseudo-orbit can be approximated by an actual orbit of a tent map with a perhaps slightly larger slope.

ABSTRACT. We study the family of tent maps-continuous, unimodal, piecewise linear maps of the interval with slopes ±s, \/2 < s < 2. We show that tent maps have the shadowing property (every pseudo-orbit can be approximated by an actual orbit) for almost all parameters s, although they fail to have the shadowing property for an uncountable, dense set of parameters.We also show that for any tent map, every pseudo-orbit can be approximated by an actual orbit of a tent map with a perhaps slightly larger slope.