31 problems

Let $A\subseteq\mathbb N$ have natural density zero. Does there exist a set $B\subseteq\mathbb N$ such that … and … as $N\to\infty$?

A question of Erdös and Nathanson is the following. Suppose $a_1 < a_2 < \ldots$ is a minimal basis which has positive density. Can it happen that for any $a_k$, the (upper) densit…

Does there exist a set $A\subseteq\mathbb{N}$ such that, as $N\to\infty$, the counting function $\lvert A\cap\{a\in\mathbb{N}:a\le N\}\rvert$ is $O(N/\log N)$, and such that for al…

Let $a_1 < a_2 < \ldots$ be any infinite sequence of integers. Straus and I conjectured that there always exists a sequence $b_1 < b_2 < \ldots$ of density $0$, so that every suffi…

Perhaps the following rather silly conjecture could be added. Is it true that the set of odd integers not of the form $2^k + p$ is the not necessarily disjoint union of an infinite…

For $r\geq 2$ let $h(r)$ be the maximal finite $k$ such that there exists a basis $A\subseteq \mathbb{N}$ of order $r$ (so every large integer is the sum of at most $r$ integers fr…

Let $A\subseteq\mathbb N$ be an asymptotic additive basis of order $2$: every sufficiently large natural number is a sum of two elements of $A$. Suppose that for every…

For a set $A$, let $A_m(x)$ denote $\left| \{ a_{i_1} + \ldots + a_{i_m} : a_{i_k} \in A \} \cap \{1, \ldots, x\} \right|$. If $A$ is a basis and $A_1(x) = o(x)$ is it true that …

Let $A=\{a_1<a_2<\cdots\}$ be such that every positive integer $n$ has a unique representation $n=a_i-a_j$ with $a_i,a_j\in A$. How slowly can the numerator $a_i=a_i^{(n)}$ grow re…

Rényi and I proved by the probabilistic method that there is a sequence $a_k < ck^r$ for which … Probably (10) holds even for a basis of order $r$. In other words there is a sequen…

Let $\alpha>0$ be irrational and set … where $\|x\|$ is the distance from $x$ to the nearest integer. Must $A_\alpha$ be an additive basis of order $2$?

Let $A,B\subseteq\mathbb N$ be infinite sets, and let $a_n$ and $b_n$ be their increasing enumerations, so that $a_n/b_n\to1$. Suppose that $A+B$ contains every sufficiently large…

Let $A\subset\mathbb{N}$ be an additive basis of order $k$ which is minimal, in the sense that if $B\subset A$ is any infinite set then $A\backslash B$ is not a basis of order $k$.…

Let $A=\{a_1<a_2<\cdots\}\subseteq\mathbb{N}$ be an asymptotic basis of order $k$, so every sufficiently large integer is a sum of at most $k$ elements of $A$. If…

Let $k\geq 3$ and $A$ be an additive basis of order $k$. Does there exist a constant $c=c(k)>0$ such that if $r(n)\geq c\log n$ for all large $n$ then $A$ must contain a minimal ba…

If $A_1,A_2$ are disjoint additive bases of order $2$ (i.e. $A_i+A_i$ contains all large integers) then must $A=A_1\cup A_2$ contain a minimal additive basis of order $2$ (one such…

For a set $A\subseteq\mathbb N$ and $o,n\in\mathbb N$, let $r_{A,o}(n)$ be the number of functions $a:\{0,\ldots,o-1\}\to\mathbb N$ such that every value of $a$ lies in $A$ and … I…

Let $A=\{a_1<\cdots\}$ be an infinite sequence of integers. Let $f(n)$ count the number of solutions to $n=\sum_{u\leq i\leq v}a_i.$ Is there such an $A$ for which…

If $A\subseteq\mathbb N$ is an additive basis of order $r$, must the integers representable as sums of $r$ distinct elements of $A$ have positive lower density?

The restricted order of a basis is the least integer $t$ (if it exists) such that every large integer is the sum of at most $t$ distinct summands from $A$. What are necessary and s…

Let $A\subset \mathbb{N}$ be an additive basis of order $2$. Must there exist $B=\{b_1<b_2<\cdots\}\subseteq A$ which is also a basis such that…

Is there $A\subseteq \mathbb{N}$ such that $\lim_{n\to \infty}\frac{1_A\ast 1_A(n)}{\log n}$ exists and is $\neq 0$?

A sharpening of our old conjecture with Turán would state: If $a_n < Cn^2$ for all $n$ then $\limsup f(n) = \infty$. In fact, for what functions $g(n) \to \infty$ does…

The sequence B is called a basis of order $k$, if every integer is the sum of $k$ or fewer $b$-s. Let $d_s(\mathrm{A}) = \alpha$ then I proved that … Thus every base is an essentia…

Let $b_1 < b_2 < \ldots$ be an infinite sequence of integers, so that every integer is of the form $b + k^2$. Denote by $B(x)$ the number $b$-s not exceeding $x$. [...] I can not d…