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 natural numbers, the sieve produces the Wythoff pair as a Beatty partition. For arithmetic progressions $a\mathbb{N}+b$, we establish a connection with the $(j,x,y,z)$-hiccup sequences recently studied by Fokkink and Joshi and with Fraenkel's complementary partitions. We further introduce an extraction sieve that also produces hiccup sequences, and whose action on arithmetic progressions is governed by an explicit affine transformation of hiccup parameters.