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 and natural number , it is easy to find the integer such that is closest to . For example, for the real number and we have . If we call the closeness of to the difference between and , 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 we can always find a sequence of values for in the set where the closeness tends to zero.

More mathematically let denote the distance from to the nearest integer then is a Heilbronn set if and only if for every real number and every there exists such that .[1]

Examples

edit

The natural numbers are a Heilbronn set as Dirichlet's approximation theorem shows that there exists   with  .

The  th powers of integers are a Heilbronn set. This follows from a result of I. M. Vinogradov who showed that for every   and   there exists an exponent   and   such that  .[2] In the case   Hans Heilbronn was able to show that   may be taken arbitrarily close to 1/2.[3] Alexandru Zaharescu has improved Heilbronn's result to show that   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

edit

The powers of 10 are not a Heilbronn set. Take   then the statement that   for some   is equivalent to saying that the decimal expansion of   has run of three zeros or three nines somewhere. This is not true for all real numbers.

References

edit
  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  ". 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  ". Invent. Math. 121 (2): 379–388. doi:10.1007/BF01884304. MR 1346212. S2CID 120435242.