Lifting with Sunflowers.

Query-to-communication lifting theorems translate lower bounds on query complexity to lower bounds for the corresponding communication model. In this paper, we give a simplified proof of deterministic lifting (in both the tree-like and dag-like settings). Whereas previous proofs used sophisticated Fourier analytic techniques, our proof uses elementary counting together with a novel connection to the sunflower lemma. In addition to a simplified proof, our approach also gives quantitative improvements in terms of gadget size. Focusing on one of the most widely used gadgets—the index gadget—existing lifting techniques are known to require at least a quadratic gadget size. Our new approach combined with robust sunflower lemmas allows us to reduce the gadget size to near linear. We conjecture that it can be further improved to poly logarithmic, similar to the known bounds for the corresponding robust sunflower lemmas.