61 research outputs found
An improved bound on the least common multiple of polynomial sequences
Cilleruelo conjectured that if of degree is
irreducible over the rationals, then
as . He
proved it for the case . Very recently, Maynard and Rudnick proved there
exists with , and showed one can take . We give an
alternative proof of this result with the improved constant . We
additionally prove the bound
and make the stronger conjecture that
as
A reverse Sidorenko inequality
Let be a graph allowing loops as well as vertex and edge weights. We
prove that, for every triangle-free graph without isolated vertices, the
weighted number of graph homomorphisms satisfies the inequality
where denotes the degree of vertex in . In particular, one has for every -regular
triangle-free . The triangle-free hypothesis on is best possible. More
generally, we prove a graphical Brascamp-Lieb type inequality, where every edge
of is assigned some two-variable function. These inequalities imply tight
upper bounds on the partition function of various statistical models such as
the Ising and Potts models, which includes independent sets and graph
colorings.
For graph colorings, corresponding to , we show that the
triangle-free hypothesis on may be dropped; this is also valid if some of
the vertices of are looped. A corollary is that among -regular graphs,
maximizes the quantity for every and ,
where counts proper -colorings of .
Finally, we show that if the edge-weight matrix of is positive
semidefinite, then This implies that among -regular graphs,
maximizes . For 2-spin Ising models, our results give a
complete characterization of extremal graphs: complete bipartite graphs
maximize the partition function of 2-spin antiferromagnetic models and cliques
maximize the partition function of ferromagnetic models.
These results settle a number of conjectures by Galvin-Tetali, Galvin, and
Cohen-Csikv\'ari-Perkins-Tetali, and provide an alternate proof to a conjecture
by Kahn.Comment: 30 page
Exponential improvements for superball packing upper bounds
We prove that for all fixed , the translative packing density of unit
-balls in is at most with
. This is the first exponential improvement in high
dimensions since van der Corput and Schaake (1936)
Local limit theorems for subgraph counts
We introduce a general framework for studying anticoncentration and local
limit theorems for random variables, including graph statistics. Our methods
involve an interplay between Fourier analysis, decoupling, hypercontractivity
of Boolean functions, and transference between ``fixed-size'' and
``independent'' models. We also adapt a notion of ``graph factors'' due to
Janson.
As a consequence, we derive a local central limit theorem for connected
subgraph counts in the Erd\H{o}s-Renyi random graph , building on work
of Gilmer and Kopparty and of Berkowitz. These results improve an
anticoncentration result of Fox, Kwan, and Sauermann and partially answers a
question of Fox, Kwan, and Sauermann. We also derive a local limit central
limit theorem for induced subgraph counts, as long as is bounded away from
a set of ``problematic'' densities, partially answering a question of Fox,
Kwan, and Sauermann. We then prove these restrictions are necessary by
exhibiting a disconnected graph for which anticoncentration for subgraph counts
at the optimal scale fails for all constant , and finding a graph for
which anticoncentration for induced subgraph counts fails in . These
counterexamples resolve anticoncentration conjectures of Fox, Kwan, and
Sauermann in the negative.
Finally, we also examine the behavior of counts of -term arithmetic
progressions in subsets of and deduce a local limit
theorem wherein the behavior is Gaussian at a global scale but has nontrivial
local oscillations (according to a Ramanujan theta function). These results
improve on results of and answer questions of the authors and Berkowitz, and
answer a question of Fox, Kwan, and Sauermann
Subgraph distributions in dense random regular graphs
Given connected graph which is not a star, we show that the number of
copies of in a dense uniformly random regular graph is asymptotically
Gaussian, which was not known even for being a triangle. This addresses a
question of McKay from the 2010 International Congress of Mathematicians. In
fact, we prove that the behavior of the variance of the number of copies of
depends in a delicate manner on the occurrence and number of cycles of length
as well as paths of length in . More generally, we provide
control of the asymptotic distribution of certain statistics of bounded degree
which are invariant under vertex permutations, including moments of the
spectrum of a random regular graph.
Our techniques are based on combining complex-analytic methods due to McKay
and Wormald used to enumerate regular graphs with the notion of graph factors
developed by Janson in the context of studying subgraph counts in
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