We prove a “special point” result for products of elliptic modular surfaces, elliptic curves, multiplicative groups and complex lines, and deduce a result about vanishing linear combinations of singular moduli and roots of unity.

We give a simple graph-theoretic proof of a classical result due to Nash-Williams on covering graphs by forests. Moreover, we derive a slight generalization of this statement where some edges are preassigned to distinct forests.

Let ${\mathrm{OT} }_{d} (n)$ be the smallest integer $N$ such that every $N$-element point sequence in ${ \mathbb{R} }^{d} $ in general position contains an order-type homogeneous subset of size $n$, where a set is order-type homogeneous if all $(d+ 1)$-tuples from this set have the same orientation. It is known that a point sequence in ${ \mathbb{R} }^{d} $ that is order-type homogeneous, forms the vertex set of a convex polytope that is combinatorially equivalent to a cyclic polytope in ${ \mathbb{R} }^{d} $. Two famous theorems of Erdős and Szekeres from 1935 imply that ${\mathrm{OT} }_{1} (n)= \Theta ({n}^{2} )$ and ${\mathrm{OT} }_{2} (n)= {2}^{\Theta (n)} $. For $d\geq 3$, we give new bounds for ${\mathrm{OT} }_{d} (n)$. In particular, we show that ${\mathrm{OT} }_{3} (n)= {2}^{{2}^{\Theta (n)} } $, answering a question of Eliáš and Matoušek, and, for $d\geq 4$, we show that ${\mathrm{OT} }_{d} (n)$ is bounded above by an exponential tower of height $d$ with $O(n)$ in the topmost exponent.

A natural number $n$ is called abundant if the sum of the proper divisors of $n$ exceeds $n$. For example, $12$ is abundant, since $1+ 2+ 3+ 4+ 6= 16$. In 1929, Bessel-Hagen asked whether or not the set of abundant numbers possesses an asymptotic density. In other words, if $A(x)$ denotes the count of abundant numbers belonging to the interval $[1, x] $, does $A(x)/ x$ tend to a limit? Four years later, Davenport answered Bessel-Hagen’s question in the affirmative. Calling this density $\Delta $, it is now known that $0. 24761\lt \Delta \lt 0. 24766$, so that just under one in four numbers are abundant. We show that $A(x)- \Delta x\lt x/ \mathrm{exp} (\mathop{(\log x)}\nolimits ^{1/ 3} )$ for all large $x$. We also study the behavior of the corresponding error term for the count of so-called $\alpha $-abundant numbers.

We solve the equation ${x}^{a} + {x}^{b} + 1= {y}^{q} $ in positive integers $x, y, a, b$ and $q$ with $a\gt b$ and $q\geq 2$ coprime to $\phi (x)$. This requires a combination of a variety of techniques from effective Diophantine approximation, including lower bounds for linear forms in complex and $p$-adic logarithms, the hypergeometric method of Thue and Siegel applied $p$-adically, local methods, and the algorithmic resolution of Thue equations.

Let $P$ and $Q$ be non-zero integers. The generalized Fibonacci sequence $\{ {U}_{n} \} $ and Lucas sequence $\{ {V}_{n} \} $ are defined by ${U}_{0} = 0$, ${U}_{1} = 1$ and ${U}_{n+ 1} = P{U}_{n} + Q{U}_{n- 1} $ for $n\geq 1$ and ${V}_{0} = 2, {V}_{1} = P$ and ${V}_{n+ 1} = P{V}_{n} + Q{V}_{n- 1} $ for $n\geq 1$, respectively. In this paper, we assume that $Q= 1$. Firstly, we determine indices $n$ such that ${V}_{n} = k{x}^{2} $ when $k\vert P$ and $P$ is odd. Then, when $P$ is odd, we show that there are no solutions of the equation ${V}_{n} = 3\square $ for $n\gt 2$. Moreover, we show that the equation ${V}_{n} = 6\square $ has no solution when $P$ is odd. Lastly, we consider the equations ${V}_{n} = 3{V}_{m} \square $ and ${V}_{n} = 6{V}_{m} \square $. It has been shown that the equation ${V}_{n} = 3{V}_{m} \square $ has a solution when $n= 3, m= 1$, and $P$ is odd. It has also been shown that the equation ${V}_{n} = 6{V}_{m} \square $ has a solution only when $n= 6$. We also solve the equations ${V}_{n} = 3\square $ and ${V}_{n} = 3{V}_{m} \square $ under some assumptions when $P$ is even.

Let $k\geq 2$ and $\Pi (n)= { \mathop{\prod }\nolimits}_{i= 1}^{k} ({a}_{i} n+ {b}_{i} )$ for some integers ${a}_{i} , {b}_{i} $ ($1\leq i\leq k$). Suppose that $\Pi (n)$ has no fixed prime divisors. Weighted sieves have shown for infinitely many integers $n$ that the number of prime factors $\Omega (\Pi (n))$ of $\Pi (n)$ is at most ${r}_{k} $, for some integer ${r}_{k} $ depending only on $k$. We use a new kind of weighted sieve to improve the possible values of ${r}_{k} $ when $k\geq 4$.

We solve a randomized version of the following open question: is there a strictly convex, bounded curve $\gamma \subset { \mathbb{R} }^{2} $ such that the number of rational points on $\gamma $, with denominator $n$, approaches infinity with $n$? Although this natural problem appears to be out of reach using current methods, we consider a probabilistic analogue using a spatial Poisson process that simulates the refined rational lattice $(1/ d){ \mathbb{Z} }^{2} $, which we call ${M}_{d} $, for each natural number $d$. The main result here is that with probability $1$ there exists a strictly convex, bounded curve $\gamma $ such that $\vert \gamma \cap {M}_{d} \vert \rightarrow + \infty , $ as $d$ tends to infinity. The methods include the notion of a generalized affine length of a convex curve as defined by F. V. Petrov [Estimates for the number of rational points on convex curves and surfaces. Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 344 (2007), 174–189; Engl. transl. J. Math. Sci. 147(6) (2007), 7218–7226].

We investigate sums of mixed powers involving two squares and three biquadrates. In particular, subject to the truth of the Generalised Riemann Hypothesis and the Elliott–Halberstam conjecture, we show that all large natural numbers $n$ with $8\nmid n$, $n\not\equiv 2~(\text{mod} ~3)$ and $n\not\equiv 14~(\text{mod} ~16)$ are the sum of two squares and three biquadrates.

A monic polynomial in ${\mathbf{F} }_{q} [t] $ of degree $n$ over a finite field ${\mathbf{F} }_{q} $ of odd characteristic can be written as the sum of two irreducible monic elements in ${\mathbf{F} }_{q} [t] $ of degrees $n$ and $n- 1$ if $q$ is larger than a bound depending only on $n$. The main tool is a sufficient condition for simultaneous primality of two polynomials in one variable $x$ with coefficients in ${\mathbf{F} }_{q} [t] $.

We give a new, geometric proof of the section conjecture for fixed points of finite group actions on projective curves of positive genus defined over the field of complex numbers, as well as its natural nilpotent analogue. As a part of our investigations we give an explicit description of the abelianised section map for groups of prime order in this setting. We also show a version of the $2$-nilpotent section conjecture.

We prove a normalized version of the restricted invertibility principle obtained by Spielman and Srivastava in [An elementary proof of the restricted invertibility theorem. Israel J. Math. 190 (2012), 83–91]. Applying this result, we get a new proof of the proportional Dvoretzky–Rogers factorization theorem recovering the best current estimate in the symmetric setting while we improve the best known result in the non-symmetric case. As a consequence, we slightly improve the estimate for the Banach–Mazur distance to the cube: the distance of every $n$-dimensional normed space from ${ \ell }_{\infty }^{n} $ is at most $\mathop{(2n)}\nolimits ^{5/ 6} $. Finally, using tools from the work of Batson et al in [Twice-Ramanujan sparsifiers. In STOC’09 – Proceedings of the 2009 ACM International Symposium on Theory of Computing, ACM (New York, 2009), 255–262], we give a new proof for a theorem of Kashin and Tzafriri [Some remarks on the restriction of operators to coordinate subspaces. Preprint, 1993] on the norm of restricted matrices.

Let $X$ be an infinite-dimensional uniformly smooth Banach space. We prove that $X$ contains an infinite equilateral set. That is, there exist a constant $\lambda \gt 0$ and an infinite sequence $\mathop{({x}_{i} )}\nolimits_{i= 1}^{\infty } \subset X$ such that $\Vert {x}_{i} - {x}_{j} \Vert = \lambda $ for all $i\not = j$.

In this paper, we prove that if $X$ is an infinite-dimensional real Hilbert space and $J: X\rightarrow \mathbb{R} $ is a sequentially weakly lower semicontinuous ${C}^{1} $ functional whose Gâteaux derivative is non-expansive, then there exists a closed ball $B$ in $X$ such that $(\mathrm{id} + {J}^{\prime } )(B)$ intersects every convex and dense subset of $X$.

We examine the minimal magnitude of perturbations necessary to change the number $N$ of static equilibrium points of a convex solid $K$. We call the normalized volume of the minimally necessary truncation robustness and we seek shapes with maximal robustness for fixed values of $N$. While the upward robustness (referring to the increase of $N$) of smooth, homogeneous convex solids is known to be zero, little is known about their downward robustness. The difficulty of the latter problem is related to the coupling (via integrals) between the geometry of the hull $\mathrm{bd} \hspace{0.167em} K$ and the location of the center of gravity $G$. Here we first investigate two simpler, decoupled problems by examining truncations of $\mathrm{bd} \hspace{0.167em} K$ with $G$ fixed, and displacements of $G$ with $\mathrm{bd} \hspace{0.167em} K$ fixed, leading to the concept of external and internal robustness, respectively. In dimension 2, we find that for any fixed number $N= 2S$, the convex solids with both maximal external and maximal internal robustness are regular $S$-gons. Based on this result we conjecture that regular polygons have maximal downward robustness also in the original, coupled problem. We also show that in the decoupled problems, three-dimensional regular polyhedra have maximal internal robustness, however, only under additional constraints. Finally, we prove results for the full problem in the case of three-dimensional solids. These results appear to explain why monostatic pebbles (with either one stable or one unstable point of equilibrium) are found so rarely in nature.