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Decidability of multiplicative matrix equations and related Diophantine problems
Sebastian Heintze, Armand Noubissie, Robert F. Tichy
https://arxiv.org/abs/2506.03932

Decidability of multiplicative matrix equations and related Diophantine problems
Some new decidability results for multiplicative matrix equations over algebraic number fields are established. In particular, special instances of the so-called knapsack problem are considered. The proofs are based on effective methods for Diophantine problems in finitely generated domains as presented in the recent book of Evertse and Györy. The focus lies on explicit bounds for the size of the solutions in terms of heights as well as on bounds for the number of solutions. This approach also…

Patterns Within the Markov Tree
Robert A. Gore
https://arxiv.org/abs/2506.04299 https://arxiv.org/pdf/2506.04299

Patterns Within the Markov Tree
An analysis of the Markov tree is presented. Markov triplets, {x,R,z}, are the positive integer solutions to the Diophantine equation x2 + R2 + z2 = 3xRz. Inspired by patterns of the Fibonacci and Pell triplets in Region 1 and Region 2 of the tree, an investigation of interior regions of the Markov tree finds generating functions and sequence functions for all triplets of all regions. These sequence functions lead to the discovery of a Pell equation for the Markov region numbers along the edges…

This https://arxiv.org/abs/2503.18789 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

On the Sylvester program and Cayley algorithm for vector partition reduction
A vector partition problem asks for a number of nonnegative integer solutions to a system of several linear Diophantine equations with integer nonnegative coefficients. J.J. Sylvester put forward an idea of reduction of vector partition to a sum of scalar partitions. In the simplest case of two equations with positive coefficients A. Cayley performed a reduction of the corresponding double partition to a sum of scalar partitions using an algorithm subject to a set of conditions on the coefficie…

This https://arxiv.org/abs/2404.04502 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

The interplay between additive and symmetric large sets and their combinatorial applications
The study of symmetric structures is a new trend in Ramsey theory. Recently in [7], Di Nasso initiated a systematic study of symmetrization of classical Ramsey theoretical results, and proved a symmetric version of several Ramsey theoretic results. In this paper Di Nasso asked if his method could be adapted to find new non-linear Diophantine equations that are partition regular [7,Final remarks (4)]. By analyzing additive, multiplicative, and symmetric large sets, we construct new partition reg…

Geometric Littlewood-Offord problems via lattice point counting
Alexandr Grebennikov, Matthew Kwan
https://arxiv.org/abs/2505.24699 https://

Geometric Littlewood-Offord problems via lattice point counting
Consider nonzero vectors $a_{1},\dots,a_{n}\in\mathbb{C}^{k}$, independent Rademacher random variables $ξ_{1},\dots,ξ_{n}$, and a set $S\subseteq\mathbb{C}^{k}$. What upper bounds can we prove on the probability that the random sum $ξ_{1}a_{1}+\dots+ξ_{n}a_{n}$ lies in $S$? We develop a general framework that allows us to reduce problems of this type to counting lattice points in $S$. We apply this framework with known results from diophantine geometry to prove various bounds when $S$ is a …