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Robin Saunders
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Sequences of evenly-distributed points in a product of intervals

Let φ be the golden ratio, (1+√5)/2. Taking the fractional parts of its integer multiples, we obtain a sequence of values in (0,1) which are in some sense "evenly distributed" in a way which is due to the continued fraction form of φ, making the constant "as difficult as possible" to approximate using rational values (otherwise, the values in the sequence would cluster around multiples of such rational approximations). If one takes the first n values, especially if n is a Fibonacci number, they will be very evenly spaced; in fact, if n is a Fibonacci number, then the difference between two consecutive values (after ordering) is always one of two adjacent powers of φ, in correspondence with the fact that the Fibonacci numbers themselves are roughly of the form φk/√5.

Is there any related (or otherwise?) sequence of values in (0,1)d, where d > 1, which are similarly "evenly distributed"?

Robin Saunders
  • 3.6k
  • 24
  • 34