Heilbronn set

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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].[1]

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].[2] In the case [math]\displaystyle{ k=2 }[/math] Hans Heilbronn was able to show that [math]\displaystyle{ \eta_2 }[/math] may be taken arbtrarily close to 1/2.[3] Alexandru Zaharescu has improved Heilbronn's result to show that [math]\displaystyle{ \eta_2 }[/math] 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 [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.

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 [math]\displaystyle{ n^2\theta\pmod 1 }[/math]". Quart. J. Math., Oxford Ser. 19: 249–256. doi:10.1093/qmath/os-19.1.249. 
  4. Zaharescu, Alexandru (1995). "Small values of [math]\displaystyle{ n^2\alpha\pmod 1 }[/math]". Invent. Math. 121 (2): 379–388. doi:10.1007/BF01884304.