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
For example, for the real number
If we call the closeness of
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.
denote the distance from
to the nearest integer then
is a Heilbronn set if and only if for every real number
‖ h θ ‖ < ε
[1] The natural numbers are a Heilbronn set as Dirichlet's approximation theorem shows that there exists
‖ q θ ‖ < ε
th powers of integers are a Heilbronn set.
This follows from a result of I. M. Vinogradov who showed that for every
there exists an exponent
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.
The powers of 10 are not a Heilbronn set.
θ ‖ < ε
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.