The differentiability of Pólya's function

✍ Scribed by Peter D Lax

• Publisher: Elsevier Science
• Year: 1973
• Tongue: English
• Weight: 304 KB
• Volume: 10
• Category: Article
• ISSN: 0001-8708
• DOI: 10.1016/0001-8708(73)90125-4

✦ Synopsis

In his paper [l] P6lya defines the following function P, mapping the interval [0, I] onto a right triangle T.

Let t be any number in the unit interval; expand it into a binary fraction: t = .d,d, ...

The n-th digit d,(t) of t is either 0 or 1.

For each t we assign a sequence of nested triangles Tl, Tz ,. . . as follows.

Divide the triangle T into two similar triangles by drawing the altitude:

T Suppose the triangle T is not isosceles; then the two triangles into which T is divided by the altitude are unequal; call the smaller of the two T, , the larger T, . Define Tl to be T, if the first digit uY1 of t is 0, T, if dl is 1. T, , T3 ,... are defined recursively, with T,-, taking the place of T and d, replacing dl . We denote the n-th triangle assigned to the number t by T,(t); clearly, the sequence T,(t) is nested, and the diameter of T,(t) tends to zero as n + co.