Related papers: Discrepancy bounds for normal numbers generated by…

Let $ (\bx(n))_{n \geq 1} $ be an $s-$dimensional Niederreiter-Xing sequence in base $b$. Let $D((\bx(n))_{n = 1}^{N})$ be the discrepancy of the sequence $ (\bx(n))_{n = 1}^{N} $. It is known that $N D((\bx(n))_{n = 1}^{N}) =O(\ln^s N)$ as…

We calculate the triple correlations for the truncated divisor sum $\lambda_{R}(n)$. The $\lambda_{R}(n)$'s behave over certain averages just as the prime counting von Mangoldt function $\Lambda(n)$ does or is conjectured to do. We also…

We consider the problem of digitalizing Euclidean segments. Specifically, we look for a constructive method to connect any two points in $\mathbb{Z}^d$. The construction must be {\em consistent} (that is, satisfy the natural extension of…

We give a simple example of an $n$-tuple of orthonormal elements in $L_2$ (actually martingale differences) bounded by a fixed constant, and hence subgaussian with a fixed constant but that are Sidon only with constant $\approx \sqrt n$.…

We prove that there exists $\alpha \in \mathbb{R}$ such that for any $N$ the dicrepancy $D_N$ of the sequence $\{\alpha n !\} ,\, 1\le n \le N$ satisfies $ D_N = O(\log N)$.

In parametric sequence alignment, optimal alignments of two sequences are computed as a function of the penalties for mismatches and spaces, producing many different optimal alignments. Here we give a 3/(2^{7/3}\pi^{2/3})n^{2/3} +O(n^{1/3}…

The discrepancy of a sequence measures how quickly it approaches a uniform distribution. Given a natural number $d$, any collection of one-dimensional so-called low discrepancy sequences $\left\{S_i:1\le i \le d\right\}$ can be concatenated…

It is a well known that, for odd $n$, the number of subsets of $\{1,2,\dots,n\}$ the sum of whose elements is divisible by $n$ equals the number of binary necklaces of length $n$. In this paper generalize this result in two directions. On…

Two popular and often applied methods to obtain two-dimensional point sets with the optimal order of $L_p$ discrepancy are digit scrambling and symmetrization. In this paper we combine these two techniques and symmetrize $b$-adic Hammersley…

For all $s \geq 1$ and $N \geq 1$ there exist sequences $(z_1,\ldots,z_N)$ in $[0,1]^s$ such that the star-discrepancy of these points can be bounded by $$D_N^*(z_1,\ldots,z_N) \leq c \frac{\sqrt{s}}{\sqrt{N}}.$$ The best known value for…

Fix a subset $I\subseteq \mathbb R_{>0}$ such that $\gamma=\inf\{ \sum_{i}n_ib_i-1>0 \mid n_i\in \mathbb Z_{\geq 0}, b_i\in I \}>0$. We give a explicit upper bound $\ell(\gamma)\in O(1/\gamma^2)$ as $\gamma\to 0$, such that for any smooth…

We prove that $\mathop{\mathbb{E}}_{m \leq M} \mathop{\mathbb{E}}_{n \leq N} \Lambda(n) \Lambda\bigl(n + \lfloor m^c \rfloor\bigr) = 1 + \rm{O}(\log^{2 - Bc} N)$, where $c > 2$ is a non-integer, $B \geq 3/c$, and $M$ is of order $N^{1/c}…

Let $(a_n)_{n \geq 1}$ be a sequence of distinct positive integers. In a recent paper Rudnick established asymptotic upper bounds for the minimal gaps of $\{a_n \alpha \bmod 1, 1 \leq n \leq N\}$ as $N \to \infty$, valid for Lebesgue-almost…

Given an integer $b\geqslant 2$ and a set $P$ of prime numbers, the set $T_P $ of Toeplitz numbers comprises all elements of $[0,b[$ whose digits $(a_n)_{n\geqslant 1}$ in the base-$b$ expansion satisfy $a_n=a_{pn}$ for all $p\in P$ and…

Let $\Omega(n)$ denote the total number of prime divisors of $n$ (counting multiplicity) and let $\omega(n)$ denote the number of distinct prime divisors of $n$. Various inequalities have been proved relating $\omega(N)$ and $\Omega(N)$…

In 1986 Lovasz, Spencer, and Vesztergombi proved a lower bound for the hereditary a discrepancy of a set system F in terms of determinants of square submatrices of the incidence matrix of F. As shown by an example of Hoffman, this bound can…

Let $N$ be an odd perfect number and let $a$ be its third largest prime divisor, $b$ be the second largest prime divisor, and $c$ be its largest prime divisor. We discuss steps towards obtaining a non-trivial upper bound on $a$, as well as…

We define a new metric between natural numbers induced by the $\ell_\infty$ norm of their unique prime signatures. In this space, we look at the natural analog of the number line and study the arithmetic function $L_\infty(N)$, which…

The necklace polynomials \[ M_n(x)=\frac1n\sum_{d\mid n}\mu(d)x^{n/d} \] play a central role in discrete mathematics: they count aperiodic necklaces, enumerate monic irreducible polynomials over finite fields, and give the dimensions of…

Define $||n||$ to be the \emph{complexity} of $n$, which is the smallest number of $1$s needed to write $n$ using an arbitrary combination of addition and multiplication. John Selfridge showed that $||n|| \geq 3\log_3 n$ for all $n$.…