The Golden Sieve and its connections to Hiccup sequences and Fraenkel games
Benoit Cloitre
Abstract
The golden sieve, a self-referential deletion process on increasing sequences of positive integers, was introduced by the author in 2002 (OEIS A099267). At each step, a pointer read from the working sequence designates which element to remove. Applied to the natural numbers, the sieve is governed by the golden ratio and recovers the Wythoff partition. We develop the theory of the golden sieve on arithmetic progressions and reveal new structural properties. The survivor sequence is a hiccup sequence in the sense of Fokkink and Joshi, whose gaps take exactly two consecutive values, selected by a rule that depends on the sequence itself. The resulting partition satisfies a Fraenkel-type complementary equation, and the gap word is Sturmian in the base case. A double sieve variant, where both pointer and target are removed at each step, yields three-way partitions of the integers. On perfect squares, the hiccup rule becomes doubly self-referential: the gap at step $n$ is controlled by whether $n$ is the square of a survivor index, so the sequence governs its own gaps through its squared values. A bootstrap argument yields the precise asymptotic growth of the survivors. We also introduce the extraction sieve $\mathcal{C}_{j,y,z}$, a family of processes that repeatedly extract and delete elements from the bottom of an infinite sequence. On $\mathbb{N}$, this sieve produces the $(j,1,y,z)$-hiccup sequence of Fokkink--Joshi, tautologically. Thus the ``silver sieve'' $\mathcal{C}_{1,3,2}$ on $\mathbb{N}$ is simply the Bosma--Dekking--Steiner $(1,1,3,2)$-hiccup with slope $1+\sqrt{2}$. On arithmetic progressions $a\mathbb{N}+b$, the sieve produces the new $(j,\,a{+}b,\,ay,\,az)$-hiccup via an affine action on parameter space, providing an algebraic bridge between sieves and hiccup sequences.