Concept

Diophantine approximation

How closely a real number can be approached by fractions with small denominators. The continued fraction supplies the best approximations and nothing else comes near, which turns questions about real numbers into questions about a short list of fractions.

Named by 2 essays across 2 fields — each of them below, with the objects they name alongside it.

$How closely a fraction can come, and the barrier that says no closer. Two panels at very different scales: the approximations to √2, which stay above the barrier a degree-two number obeys, and the truncations of a constructed number, which fall below every barrier drawn.$

Approached too fast to be algebraic

An algebraic number of degree d cannot be approached by fractions faster than the denominator's dth power. So a number that is approached faster than that is the root of no polynomial at all — and one can be built by choosing where its decimal digits go.

number · Irrationality $The only fractions that could be a cycle's shape. A table of the convergents of the base-two logarithm of three, with the approximation error, the exact value of two to the n less three to the k, and that value as a fraction of three to the k.$

How short a cycle could be

The drift argument cannot see cycles at all, which is why it is not a proof. What can see them is arithmetic — a cycle's shape has to be a fraction that approximates the logarithm of three to base two extraordinarily well, and there are very few such fractions.

dynamics · Collatz

Named alongside it

The objects these essays reach for when they reach for this one.

Continued fractions Algebraic number Collatz conjecture Construction Degree Exhaustive search Existence proof Invariant Liouville number Mean value theorem Periodic orbit Rational approximation

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