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

More mathematically let $\|\alpha\|$ denote the distance from $\alpha$ to the nearest integer then $\mathcal H$ is a Heilbronn set if and only if for every real number $\theta$ and every $\varepsilon>0$ there exists $h\in\mathcal H$ such that $\|h\theta\|<\varepsilon$ .{{cite book | first=Hugh Lowell | last=Montgomery |authorlink =Hugh Lowell Montgomery | title=Ten lectures on the Interface Between Analytic Number Theory and Harmonic Analysis| volume=84 | series=CBMS Regional Conference Series in Mathematics | year=1994 | publisher=American Mathematical Society | location=Providence Rhode Island | isbn=0-8218-0737-4 }}

Examples

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

The $k$ th powers of integers are a Heilbronn set. This follows from a result of I. M. Vinogradov who showed that for every $N$ and $k$ there exists an exponent $\eta_k>0$ and $q such that \|q^k\theta\|\ll N^{-\eta_k} . {{cite journal |first=I. M. |last=Vinogradov |authorlink= I. M. Vinogradov | title= Analytischer Beweis des Satzes uber die Verteilung der Bruchteile eines ganzen Polynoms |year= 1927| volume=21| issue=6 | pages=567–578 | journal=Bull. Acad. Sci. USSR}} In the case k=2 Hans Heilbronn was able to show that \eta_2 may be taken arbitrarily close to 1/2. {{cite journal |first=Hans|last=Heilbronn|authorlink = Hans Heilbronn |title= On the distribution of the sequence n^2\theta\pmod 1 |year= 1948| volume=19 | pages=249–256 | journal=Q. J. Math. |series=First Series |mr=0027294 | doi=10.1093/qmath/os-19.1.249}} Alexandru Zaharescu has improved Heilbronn's result to show that \eta_2 may be taken arbitrarily close to 4/7. {{cite journal |first=Alexandru |last=Zaharescu|authorlink = Alexandru Zaharescu|title= Small values of n^2\alpha\pmod 1 |year= 1995| volume=121| issue=2 | pages=379–388 | journal=Invent. Math.|mr=1346212 | doi=10.1007/BF01884304|s2cid=120435242 }}$

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 $\varepsilon=0.001$ then the statement that $\|10^k\theta\|<\varepsilon$ for some $k$ is equivalent to saying that the decimal expansion of $\theta$ has run of three zeros or three nines somewhere. This is not true for all real numbers.

References

Category:Analytic number theory

Category:Diophantine approximation