Heilbronn set

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

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

Examples
The natural numbers are a Heilbronn set as Dirichlet's approximation theorem shows that there exists $$q<[1/\varepsilon]$$ with $$\|q\theta\|<\varepsilon$$.

The $$k$$th powers of integers are a Heilbronn set. This follows from a result of I. M. Vinogradov who showed that for every $$N$$ and $$k$$ there exists an exponent $$\eta_k>0$$ and $$q<N$$ such that $$\|q^k\theta\|\ll N^{-\eta_k}$$. In the case $$k=2$$ Hans Heilbronn was able to show that $$\eta_2$$ may be taken arbitrarily close to 1/2. Alexandru Zaharescu has improved Heilbronn's result to show that $$\eta_2$$ 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 $$\varepsilon=0.001$$ then the statement that $$\|10^k\theta\|<\varepsilon$$ for some $$k$$ is equivalent to saying that the decimal expansion of $$\theta$$ has run of three zeros or three nines somewhere. This is not true for all real numbers.