Related papers: Bounds on the Minkowski constants and a function i…

Erd\H{o}s first conjectured that infinitely often we have $\varphi(n) = \sigma(m)$, where $\varphi$ is the Euler totient function and $\sigma$ is the sum of divisor function. This was proven true by Ford, Luca and Pomerance in 2010. We ask…

Let $A(s) = \sum_n a_n n^{-s}$ be a Dirichlet series with meromorphic continuation. Say we are given information on the poles of $A(s)$ with $|\Im s| \leq T$ for some large constant $T$. What is the best way to use such finite spectral data…

The Shapley-Folkman theorem shows that Minkowski averages of uniformly bounded sets tend to be convex when the number of terms in the sum becomes much larger than the ambient dimension. In optimization, Aubin and Ekeland [1976] show that…

Let $\Lambda$ be the von Mangoldt function and $r_{Q}\left(n\right)=\sum_{m_{1}+m_{2}^{2}+m_{3}^{2}=n}\Lambda\left(m_{1}\right)$ be the counting function for the numbers that can be written as sum of a prime and two squares (that we will…

One of the most fruitful results from Minkowski's geometric viewpoint on number theory is his so called 1st Fundamental Theorem. It provides an optimal upper bound for the volume of an o-symmetric convex body whose only interior lattice…

Walfisz (1963) proved the asymptotic formula \[ \sum_{n\le x}\varphi(n) = \frac{3}{\pi^2}x^2+O(x(\log x)^{\frac{2}{3}}(\log\log x)^{\frac{4}{3}}), \] which improved the error term estimate of Mertens (1874) and had been the best possible…

Let $p_n$ denote the $n$-th prime. In 2000, Panaitopol established the inequality $p_1 \cdots p_n > p_{n+1}^{n - \pi(n)}$ for all $n \geq 2$, where $\pi(x)$ is the prime counting function. In 2021, Yang and Liao refined this by introducing…

Let $m(G)$ be the infimum of the volumes of all open subgroups of a unimodular locally compact group $G$. Suppose integrable functions $\phi_1 , \phi_2 \colon G \to [0,1]$ satisfy $\| \phi_1 \| \leq \| \phi_2 \|$ and $\| \phi_1 \| + \|…

Let $\lambda_{\phi}(n)$ be the Fourier coefficients of a Hecke holomorphic or Hecke--Maass cusp form on ${\rm SL}_2(\mathbb Z)$, and $f$ be any multiplicative function that satisfies two mild hypotheses. We establish a non-trivial upper…

Tarski's theorem states that every monotone function from a complete lattice to itself has a fixed point. We analyze the query complexity of finding such a fixed point on the $k$-dimensional grid of side length $n$ under the $\leq$…

The following MW--problem was posed independently by Janusz Matkowski and Jacek Weso{\l}owski in different forms in 1985 and 2009, respectively: Are there increasing and continuous functions $\varphi\colon [0,1]\to [0,1]$, distinct from the…

In this paper we give the complete characterization of the boundedness of the generalized fractional maximal operator $$ M_{\phi,\Lambda^{\alpha}(b)}f(x) : = \sup_{Q \ni x} \frac{\|f \chi_Q\|_{\Lambda^{\alpha}(b)}}{\phi (|Q|)} \qquad (x \in…

Let f(n) denote the smallest positive integer such that every set of $f(n)$ points in general position in the Euclidean plane contains a convex n-gon. In a seminal paper published in 1935, Erd\H{o}s and Szekeres proved that f(n) exists and…

For $d \geq 2$ and i.i.d. $d$-dimensional observations $\mathbf{X}^{(1)}, \mathbf{X}^{(2)}, \ldots$ with independent Exponential$(1)$ coordinates, let $\varphi_n$ denote the minimum $\ell^1$-norm among the maxima of $\{\mathbf{X}^{(1)},…

We consider generalized Orlicz-Morrey spaces $M_{\Phi,\varphi}(\mathbb{R}^{n})$ including their weak versions $WM_{\Phi,\varphi}(\mathbb{R}^{n})$. We find the sufficient conditions on the pairs $(\varphi_{1},\varphi_{2})$ and $(\Phi, \Psi)$…

Let $(M,g)$ be a compact manifold and let $-\Delta \phi_k = \lambda_k \phi_k$ be the sequence of Laplacian eigenfunctions. We present a curious new phenomenon which, so far, we only managed to understand in a few highly specialized cases:…

Almost perfect nonlinear (APN) functions play an important role in the design of block ciphers as they offer the strongest resistance against differential cryptanalysis. Despite more than 25 years of research, only a limited number of APN…

Let $G \otimes _f H$ denote the Sierpi\'nski product of graphs $G$ and $H$ with respect to the function $f$. The Sierpi\'nski general position number ${\rm gp}{_{\rm S}}(G,H)$ is introduced as the cardinality of a largest general position…

It is shown that for any sufficiently regular even Minkowski valuation $\Phi$ which is homogeneous and intertwines rigid motions, and for any convex body $K$ in a smooth neighborhood of the unit ball, there exists a sequence of positive…

Let $f_n(k)$ be the number of subrings of index $k$ in $\mathbb{Z}^n$. We show that results of Brakenhoff imply a lower bound for the asymptotic growth of subrings in $\mathbb{Z}^n$, improving upon lower bounds given by Kaplan, Marcinek,…