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About: Heilbronn set     Goto   Sponge   NotDistinct   Permalink

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In mathematics, a Heilbronn set is an infinite set S of natural numbers for which every real number can be arbitrarily closely approximated by a fraction whose denominator is in S. For any given real number and natural number , it is easy to find the integer such that is closest to . For example, for the real number and we have . If we call the closeness of to the difference between and , the closeness is always less than 1/2 (in our example it is 0.15926...). A collection of numbers is a Heilbronn set if for any we can always find a sequence of values for in the set where the closeness tends to zero.