Browse Problems

Ruzsa suggests that a non-trivial variant of this problem arises if one imposes the stronger condition that $|A \cap \{1,\dots,N\}| \sim c_A N^{1/2}$ for some constant $c_A>0$, and similarly for $B$.

Let $A\subseteq \mathbb{N}$ and $D(A)$ be the set of those numbers which occur infinitely often as $a_1 - a_2$ with $a_1, a_2\in A$. What conditions on $A$ are sufficient to ensure $D(A)$ …

Erdos observed that this value is finite and > 1.

The smallest possible value of `lim sup n → ∞ |A ∩ {1, …, N}| / N^(1/2)` is at most `2φ^(5/2) ≈ 6.66`, with `φ` equal to the golden ratio. Proven by …

Let $A = \{1, 2, 4, 8, 13, 21, 31, 45, 66, 81, 97, \ldots\}$ be the greedy Sidon sequence: we begin with $1$ and iteratively include the next smallest integer that …

Erdős and Graham [ErGr80] also asked about the difference set $A - A$ and whether this contains $22$, which it does. [ErGr80] Erdős, P. and Graham, R., Old and new problems and …

The smallest integer which is unknown to be in $A - A$ is $33$.

It may be true that all or almost all integers are in $A - A$.

It may be true that all or almost all integers are in $A - A$.

Let $A = \{1, 2, 4, 8, 13, 21, 31, 45, 66, 81, 97, \ldots\}$ be the greedy Sidon sequence: we begin with $1$ and iteratively include the next smallest integer that …

Erdős and Graham [ErGr80] also asked about the difference set $A - A$ and whether this has positive density. [ErGr80] Erdős, P. and Graham, R., Old and new problems and results in …

Let $A=\{a_1 < \cdots < a_k\}$ be a finite set of integers and extend it to an infinite sequence $\overline{A}=\{a_1 < a_2 < \cdots \}$ by defining $a_{n+1}$ for $n \geq k$ …

Is it true that for every lacunary, strongly complete sequence `A` that is not complete whenever infinitely many terms are removed from it, `lim A (n + 1) / A n = …

Erdős and Graham [ErGr80] also say that it is not hard to construct very irregular sequences satisfying the aforementioned properties.

Erdős and Graham [ErGr80] remark that it is easy to see that if `A (n + 1) / A n > (1 + √5) / 2` then the second property is automatically …

Is there a sequence $A=\{a_1\leq a_2\leq \cdots\}$ of integers with \[\lim \frac{a_{n+1}}{a_n}=2\] such that \[P(A')= \left\{\sum_{n\in B}n : B\subseteq A'\textrm{ finite }\right\}\] has density $1$ for every cofinite subsequence $A'$ of $A$? …

For what values of $0 \leq m < n$ is there a complete sequence $A = \{a_1 \leq a_2 \leq \cdots\}$ of integers such that 1. $A$ remains complete after removing any …

For what values of $t,\alpha \in (0,\infty)$ is the sequence $\lfloor t\alpha^n\rfloor$ complete (that is, all sufficiently large integers are the sum of distinct integers of the form $\lfloor t\alpha^n\rfloor$)?

Is it true that the terms of the sequence $\lfloor (3/2)^n\rfloor$ are even infinitely often?

Is it true that the terms of the sequence $\lfloor (3/2)^n\rfloor$ are odd infinitely often and even infinitely often?

If `A ⊂ ℕ` is a finite set of integers all of whose subset sums are distinct then `∑ n ∈ A, 1/n < 2`. Proved by Ryavec.

If `A ⊂ ℕ` is a finite set of integers all of whose subset sums are distinct then `∑ n ∈ A, 1/n^s < 1/(1 - 2^(-s))`, for any `s > 0`. …

Let $p(x) \in \mathbb{Q}[x]$ be a non-constant rational polynomial with positive leading coefficient. Is it true that \[A=\{ p(n)+1/n : n \in \mathbb{N}\}\] is strongly complete, in the sense that, for any …

Let $p(x) = x \in \mathbb{Q}[x]$. It has been shown that \[A=\{ p(n)+1/n : n \in \mathbb{N}\}\] is strongly complete, in the sense that, for any finite set $B$, \[\left\{\sum_{a \in X} …

Let $p(x) = x ^ 2 \in \mathbb{Q}[x]$. It has been shown that \[A=\{ p(n)+1/n : n \in \mathbb{N}\}\] is strongly complete, in the sense that, for any finite set $B$, \[\left\{\sum_{a …

Is there some $c > 0$ such that every measurable $A \subseteq \mathbb{R}^2$ of measure $\geq c$ contains the vertices of a triangle of area 1?

Let $\alpha,\beta\in \mathbb{R}_{>0}$ such that $\alpha/\beta$ is irrational. Is \[\{ \lfloor \alpha\rfloor,\lfloor \gamma\alpha\rfloor,\lfloor \gamma^2\alpha\rfloor,\ldots\}\cup \{ \lfloor \beta\rfloor,\lfloor \gamma\beta\rfloor,\lfloor \gamma^2\beta\rfloor,\ldots\}\] complete?

Let $\alpha,\beta\in \mathbb{R}_{>0}$ such that $\alpha/\beta$ is irrational. Is \[\{ \lfloor \alpha\rfloor,\lfloor \gamma\alpha\rfloor,\lfloor \gamma^2\alpha\rfloor,\ldots\}\cup \{ \lfloor \beta\rfloor,\lfloor \gamma\beta\rfloor,\lfloor \gamma^2\beta\rfloor,\ldots\}\] complete?

Is there a lacunary sequence $A\subseteq \mathbb{N}$ (so that $A=\{a_1 < \cdots\}$ and there exists some $\lambda > 1$ such that $a_{n+1}/a_n\geq \lambda$ for all $n\geq 1$) such that \[\left\{ \sum_{a\in A'}\frac{1}{a} …

Let $f(n)$ be the maximal $k$ such that there exist integers $1 \le a_1 < \dotsc < a_k \le n$ such that all sums of the shape $\sum_{u \le i \le v} …

Let $f(n)$ be the maximal $k$ such that there exist integers $1 \le a_1 < \dotsc < a_k \le n$ such that all sums of the shape $\sum_{u \le i \le v} …

Let $f(n)$ be the maximal $k$ such that there exist integers $1 \le a_1 < \dotsc < a_k \le n$ such that all sums of the shape $\sum_{u \le i \le v} …

Let $f(n)$ be the maximal $k$ such that there exist integers $1 \le a_1 < \dotsc < a_k \le n$ such that all sums of the shape $\sum_{u \le i \le v} …

Let $f(n)$ be the maximal $k$ such that there exist integers $1 \le a_1 < \dotsc < a_k \le n$ such that all sums of the shape $\sum_{u \le i \le v} …

Let $f(n)$ be the maximal $k$ such that there exist integers $1 \le a_1 < \dotsc < a_k \le n$ such that all sums of the shape $\sum_{u \le i \le v} …

Let $g(n)$ be the maximal $k$ such that there exist integers $1 \le a_1, \dotsc, a_k \le n$ such that all sums of the shape $\sum_{u \le i \le v} a_i$ are …

Suppose $A$ is an infinite set such that all finite sums of consecutive terms of $A$ are distinct. Then it is conjectured that $A$ has density 0.

Suppose $A$ is an infinite set such that all finite sums of consecutive terms of $A$ are distinct. Then $A$ has lower density 0.

Suppose $A$ is an infinite set such that all finite sums of consecutive terms of $A$ are distinct. Then it is conjectured that the sum $\sum_k \frac{1}{a_k}$ converges.

Let $h(n)$ be the maximal $k$ such that there exist integers $1 \le a_1 \leq \dotsc \leq a_k \le n$ such that all sums of the shape $\sum_{u \le i \le v} …

Let $h(n)$ be the maximal $k$ such that there exist integers $1 \le a_1 \leq \dotsc \leq a_k \le n$ such that all sums of the shape $\sum_{u \le i \le v} …

Let $h(n)$ be the maximal $k$ such that there exist integers $1 \le a_1 \leq \dotsc \leq a_k \le n$ such that all sums of the shape $\sum_{u \le i \le v} …

Let $h(n)$ be the maximal $k$ such that there exist integers $1 \le a_1 \leq \dotsc \leq a_k \le n$ such that all sums of the shape $\sum_{u \le i \le v} …

Let $h(n)$ be the maximal $k$ such that there exist integers $1 \le a_1 \leq \dotsc \leq a_k \le n$ such that all sums of the shape $\sum_{u \le i \le v} …

Let $h(n)$ be the maximal $k$ such that there exist integers $1 \le a_1 \leq \dotsc \leq a_k \le n$ such that all sums of the shape $\sum_{u \le i \le v} …

Let $f(n)$ be the maximal $k$ such that there exist integers $1 \le a_1 < \dotsc < a_k \le n$ such that all sums of the shape $\sum_{u \le i \le v} …

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 $f(n)\to \infty$ as $n\to …

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 an $A$ such that $f(n)\geq 2$ for all large …

It is conjectured that if $A =\{a_1 < \cdots\}$ and $g$ counts the number of representations \[n=\sum_{u\leq i\leq v}a_i\] such that the sum has at least two terms, then for all $n$ …

When $A =\{a_1 < \cdots\}$ corresponds to the set of primes, it is conjectured that the $\limsup$ of the number of representations \[n=\sum_{u\leq i\leq v}a_i\] is infinite.