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

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

Examples

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

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