Segre's Theorem (Diophantine Approximation)

The set of Segre is a by Beniamino Segre named theorem from the mathematical field of number theory about the approximability irrational numbers by rational numbers . He generalizes Hurwitz's theorem , which in turn improves Dirichlet's approximation theorem .

Segre's theorem

The following statement applies to any real number : ${\displaystyle r\geq 0}$

For every irrational number there are infinitely many fully abbreviated fractions , which ${\displaystyle \alpha }$ ${\displaystyle {\tfrac {p}{q}}}$

${\displaystyle -{\frac {1}{{\sqrt {1+4r}}\cdot q^{2}}}<{\frac {p}{q}}-\alpha <{\frac {r}{{\sqrt {1+4r}}\cdot q^{2}}}}$

fulfill.

Goodness of the ceiling

For one obtains Hurwitz's theorem and it is known that the constant occurring there is sharp, so in general it cannot be replaced by a better constant. There may be better approximations for a single number . ${\displaystyle r=1}$ ${\displaystyle {\sqrt {1+4r}}={\sqrt {5}}}$ ${\displaystyle \alpha }$

For the other numbers of the form with the set of Segre provides the best possible constant. However, it is believed that for other values ​​of, the constant is not sharp. ${\displaystyle r={\frac {1}{n}}}$ ${\displaystyle n\in \mathbb {N} }$ ${\displaystyle r}$

literature

• B. Segre: Lattice points in infinite domains and asymmetric Diophantine approximations. Duke Math. J. 12, (1945). 337-365.
• Ivan Niven: On asymmetric Diophantine approximations. Michigan Math. J. 9 (1962) 121-123.
• P. Szüsz: On a theorem of Segre. Acta Arith. 23: 371-377 (1973).

Web links

• Segre's Theorem (Mathworld)

Individual evidence

1. ↑ Jing Cheng Tong: A conjecture of Segre on Diophantine approximation. Monthly Math. 112 (1991) no. 2, 141-147.