Abouabdillah's theorem

Abouabdillah's theorem refers to two distinct theorems in mathematics, proven by Moroccan mathematician Driss Abouabdillah: one in geometry and one in number theory.

Geometry

In geometry, similarities of a 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 2. Then:

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

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

Number theory

The number-theoretic theorem of Abouabdillah is about antichains in the partially ordered set E_N consisting of the positive integers in the interval [1,N], partially ordered by divisibility. With this partial order, an antichain is a set of integers within this interval, such that no member of this set is a divisor of any other member. It possible to prove using ideas related to Dilworth's theorem that the maximum number of elements in an antichain of E_2n is exactly n: there exists an antichain of this size consisting of all the numbers in the subinterval [n + 1,2n], so the maximum size of an antichain is at least n. However, there are only n odd numbers within the interval [1,2n], for each odd number c in this interval at most one number of the form 2^kc may belong to any antichain, and every number in the interval has this form for some c, so the maximum size of an antichain is also at most n.

Abouabdillah's theorem characterizes more precisely the numbers that may belong to an antichain of maximum size in E_2n. Specifically, if x is any integer in the interval [1,2n], decompose x as the product of a power of two and an odd number: x = 2^kc, where c is odd. Then, according to Abouabdillah's theorem, there exists an antichain of cardinality n in E_2n that contains x if and only if 2n < 3^k + 1c.

The smallest value in any maximum antichain of E_2n is at least 2^k, where 3^k + 1 is the first power of three that is greater than 2n, as had been posed as a problem by Paul Erdős (1937) and solved by Emma Lehmer (1939). Lehmer's solution immediately implies the special case of Abouabdillah's theorem for c = 1. Abouabdillah's theorem generalizes this solution to all values within the given interval.

References

Abouabdillah, D. (1991), "Sur les similitudes d'un espace euclidien", Revue de Mathématiques Spéciales, 7 .
Abouabdillah, D.; Turgeon, J. (1984), "On a 1937 problem of Paul Erdős concerning certain finite sequences of integers none divisible by another", Proc. 15th Southeastern Conf. Combinatorics, Graph Theory and Computing (Baton Rouge, La., 1984), Congressus Numerantium, 43, Winnipeg, Canada: Util. Math., pp. 19–22, MR 0777348 .
Erdős, Paul (1937), "Advanced Problem 3820", Problems for Solution: 3820–3823, American Mathematical Monthly, 44 (3): 179, JSTOR 2301675, doi:10.2307/2301675 .
Lehmer, Emma (1939), "Solution to Advanced Problem 3820", American Mathematical Monthly, 46 (4): 240–241, JSTOR 2303086 .

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