Abouabdillah's theorem

Abouabdillah's theorem refers to two distinct theorems in mathematics: one in geometry and one in number theory.

Geometry

In geometry, similarities of an Euclidean space preserve circles and spheres. Conversely, Abouabdillah's theorem states that every injective or surjective transformation of a Euclidean space that preserves circles or spheres is a similarity.

More precisely:

Theorem. Let $E$ be a Euclidean affine space of dimension at least 3. Then:

1. Every surjective mapping $f:\; E\; ightarrow\; E$ that transforms any four concyclic points into four concyclic points is a similarity.

2. Every injective mapping $f:\; E\; ightarrow\; E$ that transforms any circle into a circle is a similarity.

Number Theory

In number theory Abouabdillah's theorem is about antichains of N. ( An antichain of N, for divisibility, is a set of non null integers such that no one is divisible by another. It possible to prove using Dilworth's theorem that the maximal cardinality of an antichain of $E\_\{2n\}$ = {1,2,...,2n} is n).

Abouabdillah's Theorem. Let $c\; in\; E\_\{2n\}$, c = 2^kc', c' odd. Then $E\_\{2n\}$ contains an antichain of cardinality n containing c if and only if 2n < 3^k+1c'.

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*Driss Abouabdillah