Conjecture:
Given any $d \in \mathbb{Z}_{\geq 2}$ and $k=2d-2$, we have \begin{align*} \max \{ n : (\exists &f \in \{ \mathbb{Z}_{\geq 0} \to \mathbb{Z}_{\geq 0} \})\\ &[((\forall i \in \mathbb{Z}_{\geq 0}) [f(i+2) - f(i+1) \geq f(i+1) - f(i) \geq 1]) \\ & \land (S = \{f(j) : j \in [0,k-1]\}) \\ & \land (S+S \supseteq [0,n])] \} \\ = d^2 - 2.\end{align*}
Is this conjecture true?
An equivalent statement (possibly easier to understand):
Given any integer $d \geq 2$, let $k = 2d-2$. Suppose a set of integers $A_k = \{0 = a_0 < a_1 < \cdots < a_{k-1} \}$ satisfies the property that $a_{i+1} - a_i \leq a_{j+1} - a_j$ for all $0 \leq i \leq j \leq k - 2$. Also suppose that $A_k + A_k \supseteq [0,n]$, for some integer $n$. Then, $n$ is at most $d^2 - 2$.
A few notes on notation:
- $[a,b]$ denotes the integer interval $\{t \in \mathbb{Z}: a \leq t \leq b\}$, where $a,b \in \mathbb{Z}$.
- $A+A$ denotes the sumset $\{a_1+a_2 : a_1,a_2 \in A\}$.