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 (in our example it is 0.15926...). 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 } .
Examples The natural numbers are a Heilbronn set as Dirichlet's approximation theorem shows that there exists q < [ 1 / ε ] {\displaystyle q<[1/\varepsilon ]} with ‖ q θ ‖ < ε {\displaystyle \|q\theta \|<\varepsilon } . The k {\displaystyle k} th powers of integers are a Heilbronn set. This follows from a result of I. M. Vinogradov who showed that for every N {\displaystyle N} and k {\displaystyle k} there exists an exponent η k > 0 {\displaystyle \eta _{k}>0} and q < N {\displaystyle q<N} such that ‖ q k θ ‖ ≪ N − η k {\displaystyle \|q^{k}\theta \|\ll N^{-\eta _{k}}} . In the case k = 2 {\displaystyle k=2} Hans Heilbronn was able to show that η 2 {\displaystyle \eta _{2}} may be taken arbitrarily close to 1/2. Alexandru Zaharescu has improved Heilbronn's result to show that η 2 {\displaystyle \eta _{2}} may be taken arbitrarily close to 4/7. Any Van der Corput set is also a Heilbronn set.
Example of a non-Heilbronn set The powers of 10 are not a Heilbronn set. Take ε = 0.001 {\displaystyle \varepsilon =0.001} then the statement that ‖ 10 k θ ‖ < ε {\displaystyle \|10^{k}\theta \|<\varepsilon } for some k {\displaystyle k} is equivalent to saying that the decimal expansion of θ {\displaystyle \theta } has run of three zeros or three nines somewhere. This is not true for all real numbers.
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