Distractions of Shakin rings

We study, by means of embeddings of Hilbert functions, a class of rings which we call Shakin rings, i.e. quotients K[X1,…,Xn]/aK[X1,…,Xn]/a of a polynomial ring over a field K   by ideals a=L+Pa=L+P which are the sum of a piecewise lex-segment ideal L, as defined by Shakin, and a pure powers ideal P  . Our main results extend Abedelfatah's recent work on the Eisenbud–Green–Harris Conjecture, Shakin's generalization of Macaulay and Bigatti–Hulett–Pardue Theorems on Betti numbers and, when char(K)=0char(K)=0, Mermin–Murai Theorem on the Lex-Plus-Power inequality, from monomial regular sequences to a larger class of ideals. We also prove an extremality property of embeddings induced by distractions in terms of Hilbert functions of local cohomology modules.