The Golden Sieve

We revisit the golden sieve, a self-referential deletion process on increasing sequences of positive integers introduced by the author in 2002. Applied to the positive integers, the sieve recovers the Wythoff pair as a Beatty partition and yields the Fibonacci word. For arithmetic progressions, we prove a pointer–survivor identity, a two-gap law for the survivor gaps, and an affine rank identity relating survivors to deletions. This places the sieve in the framework of Fraenkel-type complementary partitions and connects it to Kimberling’s rank transform. We also show that for the positive integers both the deletion and survivor sequences are hiccup sequences in the sense of Fokkink and Joshi. For arithmetic progressions, the survivor sequence remains a hiccup sequence, while the deletion sequence satisfies a filtered self-referential two-gap law governed by the same binary control word. We then apply the same sieve to the squares, where the self-referential structure becomes doubly recursive and the survivor count admits an asymptotic expansion. We also introduce an extraction sieve, a process of a different type, which produces hiccup sequences including the Bosma–Dekking–Steiner sequence.