Related papers: Numbers with Integer Complexity Close to the Lower…

The even online Kolmogorov complexity of a string $x = x_1 x_2 \cdots x_{n}$ is the minimal length of a program that for all $i\le n/2$, on input $x_1x_3 \cdots x_{2i-1}$ outputs $x_{2i}$. The odd complexity is defined similarly. The sum of…

Let $k_r(n,\delta)$ be the minimum number of $r$-cliques in graphs with $n$ vertices and minimum degree $\delta$. We evaluate $k_r(n,\delta)$ for $\delta \leq 4n/5$ and some other cases. Moreover, we give a construction, which we conjecture…

If $$ \Delta(x) := \sum_{n\le x}c_n - Cx $$ denotes the error term in the classical Rankin-Selberg problem, then it is proved that $$ \int_0^X \Delta^4(x)\d x \ll_\epsilon X^{3+\epsilon},\quad \int_0^X \Delta_1^4(x)\d x \ll_\epsilon…

Fix a field $k$. When $\Delta$ is a simplicial complex on $n$ vertices with Stanley-Reisner ideal $I_\Delta$, we define and study an invariant called the $\textit{type defect}$ of $\Delta$. Except when $\Delta$ is of a single simplex, the…

Almost asymptotically tight lower bounds are derived for the Input/Output (I/O) complexity $IO_\mathcal{A}\left(n,M\right)$ of a general class of hybrid algorithms computing the product of two integers, each represented with $n$ digits in a…

The question of integer complexity asks about the minimal number of $1$'s that are needed to express a positive integer using only addition and multiplication (and parentheses). In this paper, we propose the notion of $l$-complexity of…

The Collatz Conjecture can be stated as: using the reduced Collatz function $C(n) = (3n+1)/2^x$ where $2^x$ is the largest power of 2 that divides $3n+1$, any odd integer $n$ will eventually reach 1 in $j$ iterations such that $C^j(n) = 1$.…

For a prime $p$, a restricted arithmetic progression in $\mathbb{F}_p^n$ is a triplet of vectors $x, x+a, x+2a$ in which the common difference $a$ is a non-zero element from $\{0,1,2\}^n$. What is the size of the largest $A\subseteq…

For fixed $m$ and $R\subseteq \{0,1,\ldots,m-1\}$, take $A$ to be the set of positive integers congruent modulo $m$ to one of the elements of $R$, and let $p_A(n)$ be the number of ways to write $n$ as a sum of elements of $A$. Nathanson…

Let $n \ge 2$ be an integer and $\alpha_1, \ldots, \alpha_n$ be non-zero algebraic numbers. Let $b_1, \ldots , b_n$ be integers with $b_n \not= 0$, and set $B = \max\{3, |b_1|, \ldots , |b_n|\}$. For $j =1, \ldots, n$, set $h^* (\alpha_j) =…

We prove that for all $k \ge 3$ and any integers $\Delta, n$ with $n \ge 2^\Delta,$ there exists a $k$-graph on $n$ vertices with maximum degree at most $\Delta$ such that $r(H)\geq\tw_{k-1}(c_k \Delta) \cdot n$ for some constant $c_k > 0$,…

By extending a construction due to Gross and McMullen [2], we show that for any odd integer n and for any even integer d>n+2 there are infinitely many Salem numbers $\alpha$ of degree d such that $\alpha^n-1$ is a unit. A similar result is…

Shallit and Wang showed that the automatic complexity $A(x)$ satisfies $A(x)\ge n/13$ for almost all $x\in{\{\mathtt{0},\mathtt{1}\}}^n$. They also stated that Holger Petersen had informed them that the constant 13 can be reduced to 7. Here…

We consider the complexity for computing the approximate sum $a_1+a_2+...+a_n$ of a sorted list of numbers $a_1\le a_2\le ...\le a_n$. We show an algorithm that computes an $(1+\epsilon)$-approximation for the sum of a sorted list of…

Professor Georges Rhin considers a nonzero algebraic integer $\a$ with conjugates $\a_1=\a, \ldots, \a_d$ and asks what can be said about $\d \sum_{ | \a_i | >1} | \a_i |$, that we denote ${\rm{R}}(\a)$. If $\a$ is supposed to be a totally…

For positive integers $n\geq k\geq t$, a collection $ \mathcal{B} $ of $k$-subsets of an $n$-set $ X $ is called a $t$-packing if every $t$-subset of $ X $ appears in at most one set in $\mathcal{B}$. In this paper, we give some upper and…

We establish a sharp lower-bound for the number of conjugacy classes $k(A_n)$ in the alternating group $A_n$, for $n \geq 3$. Namely, we show that $k\left(A_n\right) \geq \frac{k\left(A_7\right)}{\log_2\left|A_7\right|} \cdot…

The dimension of a partially-ordered set $P$ is the smallest integer $d$ such that one can embed $P$ into a product of $d$ linear orders. We prove that the dimension of the divisibility order on the interval $\{1, \dotsc, n\}$ is bounded…

Let $p_n$ denote the $n$-th prime. For any $m\geq 1$, there exist infinitely many $n$ such that $p_{n}-p_{n-m}\leq C_m$ for some large constant $C_m>0$, and $$p_{n+1}-p_n\geq \frac{c_m\log n\log\log n\log\log\log\log n}{\log\log\log n}, $$…

We initiate the study of Ramsey numbers of trails. Let $k \geq 2$ be a positive integer. The Ramsey number of trails with $k$ vertices is defined as the the smallest number $n$ such that for every graph $H$ with $n$ vertices, $H$ or the…