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 [math]\displaystyle{ \theta }[/math] and natural number [math]\displaystyle{ h }[/math], it is easy to find the integer [math]\displaystyle{ g }[/math] such that [math]\displaystyle{ g/h }[/math] is closest to [math]\displaystyle{ \theta }[/math]. For example, for the real number [math]\displaystyle{ \pi }[/math] and [math]\displaystyle{ h=100 }[/math] we have [math]\displaystyle{ g=314 }[/math]. If we call the closeness of [math]\displaystyle{ \theta }[/math] to [math]\displaystyle{ g/h }[/math] the difference between [math]\displaystyle{ h\theta }[/math] and [math]\displaystyle{ g }[/math], 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 [math]\displaystyle{ \theta }[/math] we can always find a sequence of values for [math]\displaystyle{ h }[/math] in the set where the closeness tends to zero. More mathematically let [math]\displaystyle{ \|\alpha\| }[/math] denote the distance from [math]\displaystyle{ \alpha }[/math] to the nearest integer then [math]\displaystyle{ \mathcal H }[/math] is a Heilbronn set if and only if for every real number [math]\displaystyle{ \theta }[/math] and every [math]\displaystyle{ \varepsilon\gt 0 }[/math] there exists [math]\displaystyle{ h\in\mathcal H }[/math] such that [math]\displaystyle{ \|h\theta\|\lt \varepsilon }[/math].

Examples

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

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