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 ${\displaystyle \theta }$ and natural number ${\displaystyle h}$, it is easy to find the integer ${\displaystyle g}$ such that ${\displaystyle g/h}$ is closest to ${\displaystyle \theta }$. For example, for the real number ${\displaystyle \pi }$ and ${\displaystyle h=100}$ we have ${\displaystyle g=314}$. If we call the closeness of ${\displaystyle \theta }$ to ${\displaystyle g/h}$ the difference between ${\displaystyle h\theta }$ and ${\displaystyle g}$, 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 ${\displaystyle \theta }$ we can always find a sequence of values for ${\displaystyle h}$ in the set where the closeness tends to zero.

More mathematically let ${\displaystyle \|\alpha \|}$ denote the distance from ${\displaystyle \alpha }$ to the nearest integer then ${\displaystyle {\mathcal {H}}}$ is a Heilbronn set if and only if for every real number ${\displaystyle \theta }$ and every ${\displaystyle \varepsilon >0}$ there exists ${\displaystyle h\in {\mathcal {H}}}$ such that ${\displaystyle \|h\theta \|<\varepsilon }$.^[1]