Open-problem statements, with their sources attached.

For $k>2$ it is not known if $f_{k,k}(x)=o(x)$.

Does there exist a polynomial $f(x)\in\mathbb{Z}[x]$ such that all the sums $f(a)+f(b)$ with $a < b$ nonnegative integers are distinct?

Probably $f(x) = x^5$ has the property that the sums $f(a)+f(b)$ with $a < b$ nonnegative integers are distinct.

Writing $f_{k, 3}(x)$ for the number of integers $\leq x$ which are the sum of three $k$th powers, is it true that $f_{k, 3}(x) \gg x ^ (3 / k)$?

Writing $f_{k, 3}(x)$ for the number of integers $\leq x$ which are the sum of three $k$th powers, is it even true that $f_{k, 3}(x) \gg_{\epsilon} x ^ (3 / k - \epsilon)$?

Let A ⊆ ℕ be a set such that every integer can be written as n^2 + a for some a in A and n ≥ 0. What is the smallest possible value of lim sup n → ∞ |A ∩ {1, …, N}| / N^(1/2)?

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 $D(A)$ has bounded gaps?

This is formalised here using the answer(sorry) mechanism. In order to solve this problem one has to provide what the sufficient conditions are, and proof that they imply the desired condition. If the condition is a solution to the problem is up to human judgement.

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 $n \geq k$ to be the least integer exceeding $a_n$ which is not of the form $a_i + a_j$ with $i,j \leq n$. Is it true that the sequence of differences $a_{m+1}-a_m$ is eventually periodic?

This problem is discussed under Problem 7 on Green's open problems list.

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 = (1 + √5) / 2?

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 $m$ elements, but
2. $A$ is not complete after removing any $n$ elements.

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 $\lfloor t\alpha^n\rfloor$)?

It seems likely that the sequence is complete for all for all $t>0$ and all $1 < \alpha < \frac{1+\sqrt{5}}{2}$.

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

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

Let $\alpha,\beta\in \mathbb{R}_{>0}$ such that $\alpha/\beta$ is irrational. Is

Let $\alpha,\beta\in \mathbb{R}_{>0}$ such that $\alpha/\beta$ is irrational. Is

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 $\sum_{u \le i \le v} a_i$ are distinct. Is $f(n)=o(n)$?

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 $\sum_{u \le i \le v} a_i$ are distinct. How does $f(n)$ grow? Can we find a (good) explicit function $g$ such that $g = O(f)$ ?

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 $\sum_{u \le i \le v} a_i$ are distinct. How does $f(n)$ grow? Can we find a (good) explicit function $g$ such that $f = O(g)$ ?