The Golden Sieve

TL;DR

The paper explores the properties of the golden sieve, a self-referential process on integers, linking it to known sequences and introducing a new extraction sieve with affine transformation properties.

Contribution

It revisits the golden sieve, connects it to hiccup sequences and Fraenkel's partitions, and introduces a new extraction sieve with explicit affine transformation rules.

Findings

The golden sieve produces the Wythoff pair as a Beatty partition.

Connections established between the sieve and hiccup sequences for arithmetic progressions.

Introduction of an extraction sieve with affine transformation governing its action.

Abstract

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.