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 ${\displaystyle \varepsilon >0}$ there exists ${\displaystyle h\in {\mathcal {H}}}$ such that ${\displaystyle \|h\theta \|<\varepsilon }$.^[1]

Examples

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

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 ${\displaystyle \eta _{k}>0}$ and ${\displaystyle q<N}$ 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 arbitrarily 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 ${\displaystyle \|10^{k}\theta \|<\varepsilon }$ 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