The Hilbert additive group scheme arises in algebraic geometry as the "unipotent completion" of the integers over Spec(Z). At the level of functions, it carries a canonical filtration compatible with the group structure. This is intimately related to the HKR filtration on Hochschild homology. 

I will review the above story and then discuss ongoing work (joint with Alice Hedenlund) on lifting the Hilbert additive group, together with its filtration on functions, to the setting of derived algebraic geometry over the sphere spectrum. This crucially uses the yoga of even filtrations, introduced by Hahn-Raksit-Wilson, which I will briefly review as well.