Two block gluing constructions

We prove two existence results about block gluing in two-dimensional SFTs. First, a large class of functions between exponential and logarithmic can be realized as block gluing functions of two-dimensional SFTs. Second, there exists an aperiodic linearly block gluing two-dimensional SFT with entropy dimension 1. These solve respective open questions of Chandgotia, Gangloff, Hellouin de Menibus and Oprocha, and Gangloff and Sablik.