This work is devoted to proving that, given an integer $x\ge 2$, there are infinitely many perfect powers coprime with $x$ having exactly $k\ge 3$ nonzero digits in their base $x$ representation, except for the case $x=2$, $k=4$, for which a known finiteness result by Corvaja and Zannier holds.

We provide a general framework on the coefficients of the graph polynomials of graphs which are Cartesian products. As a corollary, we prove that if $G=\left(V,E\right)$ is a graph with degrees of vertices $2d\left(v\right)$, $v\in V$, and the graph polynomial ${\prod }_{\left(i,j\right)\in E}\left(x_j-x_i\right)$ contains an “almost central” monomial (that is a monomial ${\prod }_v{x}_v^{c_v}$, where $|c_v-d\left(v\right)|\le 1$ for all $v\in V$), then the Cartesian product $G\square C_{2n}$ is $\left(d\left(\cdot \right)+2\right)$-choosable.

An $m$-dimensional simplex $\mathrm{\Delta }$ in ${ℝ}^m$ is called empty lattice simplex if $\mathrm{\Delta }\cap {\mathbb{Z}}^m$ is exactly the set of vertices of $\mathrm{\Delta }$. A theorem of White states that if $m=3$ then, up to an affine unimodular transformation of the lattice ${\mathbb{Z}}^m$, any empty lattice simplex $\mathrm{\Delta }\subset {ℝ}^3$ is isomorphic to a tetrahedron whose vertices have third coordinate $0$ or $1$. We prove a generalization of this theorem for some special empty lattice simplices of arbitrary odd dimension $m=2d-1$ which was conjectured by Sebő and Borisov. Our result implies a classification of all $2d$-dimensional isolated Gorenstein cyclic quotient singularities with minimal $log$-discrepancy $\ge d$.

The original knapsack problem is well known to be NP-complete. In a multidimensional version, one have to decide whether a $p\in {\mathbb{N}}^k$ is in the sumset-sum of a set $X\subseteq {\mathbb{N}}^k$. In this paper, we are going to investigate a communication complexity problem related to this. We are also going to prove some results about the special case of the multidimensional knapsack problem, when the set $X$ is in the form $X=A_1\times \cdots \times A_k\subseteq {\mathbb{N}}^k$, where $A_i$ is a so-called regular set for every $i=1,2,\dots ,k$.

Cotangent sums play a significant role in the Nyman–Beurling criterion for the Riemann hypothesis. Here we investigate the maximum of the values of these cotangent sums over various sets of rational numbers in short intervals.

We study the validity of the first-order zero-one law for the binomial $k$-partite random graph in two settings: dense (the probability $p$ of appearance of an edge is a constant) and sparse ($p=n^{-\alpha }$, where $n$ is the cardinality of each part of the graph). On the way, we prove that, for every rational $\rho \ge 1$, there exists a bipartite strictly balanced graph with density $\rho$.

We correct an error in the statement of Theorem 1.1 in Mosc. J. Comb. Number Theory 10:3 (2021), 249–260.