Open-problem statements, with their sources attached.

$W(3, 38) > 1378$ from [AKS14, Table 3].

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

$W(3, 39) > 1418$ from [AKS14, Table 3].

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 $f(n)=o(n)$?

Does there exist a Lipschitz function $f : \mathbb{N} \to \mathbb{Z}$ whose graph $\Gamma = \{(n, f(n)) : n \in \mathbb{N}\} \subseteq \mathbb{Z}^2$ is free of 3-term progressions?

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 $g = O(f)$ ?

What is the largest subset of $[N]$ with no solution to $x + 3y = 2z + 2w$ in distinct integers $x, y, z, w$?

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 $f = O(g)$ ?

From [Ruzsa] $f(N) \gg N^{1/2}$.

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 $f = \Theta(g)$ ?

From [Schoen and Sisask] $f(N) \ll N \cdot e^{-c(\log N)^{1/7}}$.

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 $g = o(f)$ ?

$f(N) \gg N \cdot e^{-c(\log N)^{1/7}}$.

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 $f = o(g)$ ?

From [Yufei Zhao]: Is there a subset of $\{1, \ldots, N\}$ of size $N^{1/3 - o(1)}$ with no nontrivial solutions to $x + 2y + 3z = x' + 2y' + 3z'$?

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 that $G$ is a finite group, and let $A \subset G \times G$ be a subset of density $\alpha$. Is it true that there are $\gg_\alpha |G|^3$ triples $x, y, g$ such that $(x, y), (gx, y), (x, gy)$ all lie in $A$?

Note: A is taken as $\alpha$-dense, i.e. $|A| \ge \alpha |G|^2$ [Au16, Question 2]

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.

[Ma21] showed that $3.13 \leq C$.

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 distinct. It is known that $\left(\frac 1 3 + o(1) \right)n \leq g(n) \leq \left(\frac 2 3 + o(1) \right)n.$