Irrationality exponent of π

For a real numberx, the irrationality index (x) is a way of measuring how wellx can be approximated by rational numbers. Ifx is rational, (x) = 1. Ifx is irrational, (x) >= 2.

OpenAI recently published a proof that () = 2. Almost all real numbers have irrationality exponent 2, so the new result says is typical in this regard. There are numbers proven to have irrationality index greater than 2 (more on that below), but isn't one of them.

The irrationality exponent (x) is defined as the supremum of the set of values such that

for infinitely many coprime integersp andq withq > 0.

This means that the approximation error for approximating with a rational numberp/q is typically on the order of 1/q^2, just like most irrational numbers.

There are numbers with higher irrationality exponents. For example, Cahen's constantC has irrationality exponent 3. This meansC is an irrational number that has infinitely many rational approximationsp/q with error less than 1/q^3.

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