OPEN
We call $A\subset \mathbb{N}$ dissociated if $\sum_{n\in X}n\neq \sum_{m\in Y}m$ for all finite $X,Y\subset A$ with $X\neq Y$.
Let $A\subset \mathbb{N}$ be an infinite set. We call $A$ proportionately dissociated if every finite $B\subset A$ contains a dissociated set of size $\gg \lvert B\rvert$.
Is every proportionately dissociated set the union of a finite number of dissociated sets?
This question appears in a paper of Alon and Erdős
[AlEr85], although the general topic was first considered by Pisier
[Pi83], who observed that the converse holds, and proved that being proportionately dissociated is equivalent to being a 'Sidon set' in the harmonic analysis sense; that is, whenever $f:A\to \mathbb{C}$ there exists some $\theta\in [0,1]$ such that
\[\| f\|_1 \ll \left\lvert\sum_{n\in A} f(n)e(n\theta)\right\rvert,\]
where $e(x)=e^{2\pi ix}$.
Alon and Erdős write that it 'seems unlikely that [this] is also sufficient'. They also point out the same question can be asked replacing dissociated with Sidon (in the additive combinatorial sense).
