# 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 $\displaystyle{ \theta }$ 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 $\displaystyle{ \varepsilon\gt 0 }$ there exists $\displaystyle{ h\in\mathcal H }$ such that $\displaystyle{ \|h\theta\|\lt \varepsilon }$.[1]

## Examples

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

The $\displaystyle{ k }$th powers of integers are a Heilbronn set. This follows from a result of I. M. Vinogradov who showed that for every $\displaystyle{ N }$ and $\displaystyle{ k }$ there exists an exponent $\displaystyle{ \eta_k\gt 0 }$ and $\displaystyle{ q\lt N }$ such that $\displaystyle{ \|q^k\theta\|\ll N^{-\eta_k} }$.[2] In the case $\displaystyle{ k=2 }$ Hans Heilbronn was able to show that $\displaystyle{ \eta_2 }$ may be taken arbtrarily close to 1/2.[3] Alexandru Zaharescu has improved Heilbronn's result to show that $\displaystyle{ \eta_2 }$ may be taken arbitrarily close to 4/7.[4]

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 $\displaystyle{ \varepsilon=0.001 }$ then the statement that $\displaystyle{ \|10^k\theta\|\lt \varepsilon }$ for some $\displaystyle{ k }$ is equivalent to saying that the decimal expansion of $\displaystyle{ \theta }$ has run of three zeros or three nines somewhere. This is not true for all real numbers.

## References

1. Montgomery, Hugh Lowell (1994). Ten lectures on the Interface Between Analytic Number Theory and Harmonic Analysis. CBMS Regional Conference Series in Mathematics. 84. Providence Rhode Island: American Mathematical Society. ISBN 0-8218-0737-4.
2. Vinogradov, I. M. (1927). "Analytischer Beweis des Satzes uber die Verteilung der Bruchteile eines ganzen Polynoms". Bull. Acad. Sci. USSR 21 (6): 567–578.
3. Heilbronn, Hans (1948). "On the distribution of the sequence $\displaystyle{ n^2\theta\pmod 1 }$". Quart. J. Math., Oxford Ser. 19: 249–256. doi:10.1093/qmath/os-19.1.249.
4. Zaharescu, Alexandru (1995). "Small values of $\displaystyle{ n^2\alpha\pmod 1 }$". Invent. Math. 121 (2): 379–388. doi:10.1007/BF01884304.