Dmitrii Zhelezov

Improved bounds for arithmetic progressions in product sets

Let B be a set of natural numbers of size n. We prove that the length of the longest arithmetic progression contained in the product set B.B = {bb′|b, b′ ∈ B} cannot be greater than O(n log n) which matches the lower bound provided in an earlier paper up to a multiplicative constant. For sets of complex numbers, we improve the bound to Oϵ(n1 + ϵ) for arbitrary ϵ > 0 assuming the GRH.

On the Diameters of Commuting Graphs Arising from Random Skew-Symmetric Matrices

We present a two-parameter family of finite, non-abelian random groups and propose that, for each fixed k, as m → ∞ the commuting graph of G_{m,k} is almost surely connected and of diameter k. We present heuristic arguments in favour of this conjecture, following the lines of classical arguments for the Erdős–Renyi random graph. As well as being of independent interest, our groups would, if our conjecture is true, provide a large family of counterexamples to the conjecture of Iranmanesh and Jafarzadeh that the commuting graph of a finite group, if connected, must have a bounded diameter. Simulations of our model yielded explicit examples of groups whose commuting graphs have all diameters from 2 up to 10.

Convex sequences may have thin additive bases

For a fixed

When can multi-agent rendezvous be executed in time linear in the diameter of a plane configuration?

c > 0

A bound on the multiplicative energy of a sum set and extremal sum-product problems

we construct an arbitrarily large set

On Additive Bases of Sets with Small Product Set

B

On additive shifts of multiplicative almost-subgroups in finite fields

of size

Can connected commuting graphs of finite groups have arbitrarily large diameter

n

Product sets cannot contain long arithmetic progressions

such that its sum set

Discrete spheres and arithmetic progressions in product sets

B+B