Related papers: On convex equations
We show that several versions of Floyd and Rivest's improved algorithm Select for finding the $k$th smallest of $n$ elements require at most $n+\min\{k,n-k\}+O(n^{1/2}\ln^{1/2}n)$ comparisons on average and with high probability. This…
In recent years, significant progress has been made in explaining the apparent hardness of improving upon the naive solutions for many fundamental polynomially solvable problems. This progress has come in the form of conditional lower…
Let $\|n\|$ stand for the integer complexity of the number $n$, i.e. for the least number of $1$'s needed to write $n$ using arbitrary many additions, multiplications, and parentheses. The two-sided inequality $3\log_3 n\leq\|n\|\leq…
We provide upper bounds on the largest subsets of $\{1,2,\dots,N\}$ with no differences of the form $h_1(n_1)+\cdots+h_{\ell}(n_{\ell})$ with $n_i\in \mathbb{N}$ or $h_1(p_1)+\cdots+h_{\ell}(p_{\ell})$ with $p_i$ prime, where $h_i\in…
We establish new quantitative estimates for general systems of functions with wavelet-type dyadic structure. These estimates are applied to obtain the optimal growth of various types of Weyl multipliers for certain wavelet-type systems.…
We develop recent ideas of Elsholtz, Proske, and Sauermann to construct denser subsets of $\{1,\dots,N\}$ that lack arithmetic progressions of length $3$. This gives the first quasipolynomial improvement since the original construction of…
This article considers the minimal non-zero (= indecomposable) solutions of the linear congruence $1\cdot x_1 + \cdots + (m-1)\cdot x_{m-1} \equiv 0 \pmod m$ for unknown non-negative integers $x_1, \ldots, x_n$, and characterizes the…
Erd\H{o}s showed that every set of $n$ positive integers contains a subset of size at least $n/(k+1)$ containing no solutions to $x_1 + \cdots + x_k = y$. We prove that the constant $1/(k+1)$ here is best possible by showing that if $(F_m)$…
We study connections between the problem of the existence of positive solutions for certain nonlinear equations and weighted norm inequalities. In particular, we obtain explicit criteria for the solvability of the Dirichlet problem…
We give a sharpened form of Siegel Lemma's w. r. t. the maximum norm. This implies a new lower bound on the greatest element of a sum-distinct set of positive integers (Erd\"os-Moser problem). The main tools are Minkowski's theorem on…
In this paper, among other things, we show that, given $r\in N$, there is a constant $c=c(r)$ such that if $f\in C^r[-1,1]$ is convex, then there is a number ${\mathcal N}={\mathcal N}(f,r)$, depending on $f$ and $r$, such that for…
The Erd\"{o}s-Straus conjecture states that the equation $\frac{4}{n}=\frac{1}{x}+\frac{1}{y}+\frac{1}{z}$ has positive integer solutions $x, y, z$ for every postive integers $n\ge 2$. We generalize the Erd\"{o}s-Straus equation, state…
We are concerned with the half-space Dirichlet problem \[\begin{array}{ll} -\Delta v+v=|v|^{p-1}v & \textrm{in}\ \mathbb{R}^N_+, v=c\ \textrm{on}\ \partial\mathbb{R}^N_+, &\lim_{x_N\to \infty}v(x',x_N)=0\ \textrm{uniformly in}\…
The discrete acyclic convolution computes the 2n-1 sums sum_{i+j=k; (i,j) in [0,1,2,...,n-1]^2} (a_i b_j) in O(n log n) time. By using suitable offsets and setting some of the variables to zero, this method provides a tool to calculate all…
It is shown, subject to the abc-conjecture, that \[\sum_{n\le N}\exp(2\pi i\alpha n^3)\ll_{\epsilon,\alpha}N^{5/7+\epsilon}\] for any $\epsilon>0$ and any quadratic irrational $\alpha$.
Let $T(n)=\left\{\begin{array}{ll}3n+1&(n\hbox{ odd})\frac n2&(n\hbox{ even})\end{array}\right.$ ($n\in\mathbb Z$). We call "the orbit of the integer $n$", the set $$ \mathcal O_n:=\{m\in\mathbb Z\;:\;\exists k\ge0,\ m=T^k(n)\} $$ and we…
A famous conjecture of Graham asserts that every set $A \subseteq \mathbb{Z}_p \setminus \{0\}$ can be ordered so that all partial sums are distinct. Although this conjecture was recently proved for sufficiently large primes by Pham and…
Within the scope of elementary number theory, we prove that, as the main result, if $1 \leq x < y < z$ are integers such that at least one of $y, z, x+y$ is prime then $x^{n}+y^{n} \neq z^{n}$ for every odd integer $n \geq 3$. This result…
We show that for any sequence $f: {\bf N} \to \{-1,+1\}$ taking values in $\{-1,+1\}$, the discrepancy $$ \sup_{n,d \in {\bf N}} \left|\sum_{j=1}^n f(jd)\right| $$ of $f$ is infinite. This answers a question of Erd\H{o}s. In fact the…
Exploring the Collatz Conjecture and changing the expression from 3n + 1 to 5n + 1, we found patterns in different sets of numbers. Some numbers reduce to one (as stated in the Collatz Conjecture), some might escape to infinity, and some…