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

More mathematically let  ‖ α ‖ {\displaystyle \|\alpha \|} denote the distance from  α {\displaystyle \alpha } to the nearest integer then  H {\displaystyle {\mathcal {H}}} is a Heilbronn set if and only if for every real number  θ {\displaystyle \theta } and every  ε > 0 {\displaystyle \varepsilon >0} there exists  h ∈ H {\displaystyle h\in {\mathcal {H}}} such that  ‖ h θ ‖ < ε {\displaystyle \|h\theta \|<\varepsilon } .[1]

## Examples

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

The  k {\displaystyle k} th powers of integers are a Heilbronn set. This follows from a result of I. M. Vinogradov who showed that for every  N {\displaystyle N} and  k {\displaystyle k} there exists an exponent  η k > 0 {\displaystyle \eta _{k}>0} and  q < N {\displaystyle q<N} such that  ‖ q k θ ‖ ≪ N − η k {\displaystyle \|q^{k}\theta \|\ll N^{-\eta _{k}}} .[2] In the case  k = 2 {\displaystyle k=2} Hans Heilbronn was able to show that  η 2 {\displaystyle \eta _{2}} may be taken arbitrarily close to 1/2.[3] Alexandru Zaharescu has improved Heilbronn's result to show that  η 2 {\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

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The powers of 10 are not a Heilbronn set. Take  ε = 0.001 {\displaystyle \varepsilon =0.001} then the statement that  ‖ 10 k θ ‖ < ε {\displaystyle \|10^{k}\theta \|<\varepsilon } for some  k {\displaystyle k} is equivalent to saying that the decimal expansion of  θ {\displaystyle \theta } has run of three zeros or three nines somewhere. This is not true for all real numbers.

## References

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1. ^ Montgomery, Hugh Lowell (1994). Ten lectures on the Interface Between Analytic Number Theory and Harmonic Analysis. CBMS Regional Conference Series in Mathematics. Vol. 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  n 2 θ ( mod 1 ) {\displaystyle n^{2}\theta {\pmod {1}}} ". Q. J. Math. First Series. 19: 249–256\. doi:10.1093/qmath/os-19.1.249. MR 0027294.
4. ^ Zaharescu, Alexandru (1995). "Small values of  n 2 α ( mod 1 ) {\displaystyle n^{2}\alpha {\pmod {1}}} ". Invent. Math. 121 (2): 379–388\. doi:10.1007/BF01884304. MR 1346212. S2CID 120435242.