Following a growing number of studies that, over the past 15 years, have established entropy inequalities via ideas and tools from additive combinatorics, in this work we obtain a number of new bounds for the differential entropy of sums, products, and sum-product combinations of continuous random variables. Partly motivated by recent work by Goh on the discrete entropic version of the notion of “additive energy”, we introduce the additive energy of pairs of continuous random variables and prove various versions of the statement that “the additive energy is large if and only if the entropy of the sum is small”, along with a version of the Balog–Szemerédi–Gowers theorem for differential entropy. Then, motivated in part by recent work by M´athé and O’Regan, we establish a series of new differential entropy inequalities for products and sum-product combinations of continuous random variables. In particular, we prove a new, general, ring Plüunnecke–Ruzsa entropy inequality. We briefly return to the case of discrete entropy and provide a characterization of discrete random variables with “large doubling”, analogous to Tao’s Freiman-type inverse sumset theory for the case of small doubling. Finally, we consider the natural entropic analog of the Erdős–Szemerédi sum-product phenomenon for integer-valued random variables. We show that, if it does hold, then the range of parameters for which it does would necessarily be significantly more restricted than its anticipated combinatorial counterpart.

Abstract. Defant, Li, Propp, and Young recently resolved two enumerative conjectures of Propp concerning the tilings of regions in the hexagonal grid called benzels using two types of prototiles called stones and bones (with varying constraints on allowed orientations of the tiles). Their primary tool, a bijection called compression that converts certain [Formula: see text]-ribbon tilings to [Formula: see text]-ribbon tilings, allowed them to reduce their problems to the enumeration of dimers (i.e., perfect matchings) of certain graphs. We present a generalized version of compression that no longer relies on the perspective of partitions and skew shapes. Using this strengthened tool, we resolve three more of Propp’s conjectures and recast several others as problems about perfect matchings.

Abstract The Delsarte linear program is used to bound the size of codes given their block length n and minimal distance d by taking a linear relaxation from codes to quasicodes. We study for which values of (n, d) this linear program has a unique optimum: while we show that it does not always have a unique optimum, we prove that it does if $$d>n/2$$ d > n / 2 or if $$d \le 2$$ d ≤ 2 . Introducing the Krawtchouk decomposition of a quasicode, we prove there exist optima to the (n, 2e) and $$(n-1,2e-1)$$ ( n - 1 , 2 e - 1 ) linear programs that have essentially identical Krawtchouk decompositions, revealing a parity phenomenon among the Delsarte linear programs. We generalize the notion of extending and puncturing codes to quasicodes, from which we see that this parity relationship is given by extending/puncturing. We further characterize these pairs of optima, in particular demonstrating that they exhibit a symmetry property, effectively halving the number of decision variables.