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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 θ {\displaystyle \theta } and natural number h {\displaystyle h} , it is easy to find the integer g {\displaystyle g} such that g / h {\displaystyle g/h} is closest to θ {\displaystyle \theta }. For example, for the real number π {\displaystyle \pi } and h = 100 {\displaystyle h=100} we have g = 314 {\displaystyle g=314}. If we call the closeness of θ {\displaystyle \theta } to g / h {\displaystyle g/h} the difference between h θ {\displaystyle h\theta } and g {\displaystyle g} , the closeness is always less than 1/2. A collection of numbers is a Heilbronn set if for any θ {\displaystyle \theta } we can always find a sequence of values for h {\displaystyle h} in the set where the closeness tends to zero.
More mathematically let ‖ α ‖ {\displaystyle \|\alpha \|} denote the distance from α {\displaystyle \alpha } to the nearest integer then H {\displaystyle {\mathcal {H}}} is a Heilbronn set if and only if for every real number θ {\displaystyle \theta } and every ε > 0 {\displaystyle \varepsilon >0} there exists h ∈ H {\displaystyle h\in {\mathcal {H}}} such that ‖ h θ ‖ < ε {\displaystyle \|h\theta \|<\varepsilon }.