The Greedy Algorithm for Dissociated Sets

TL;DR

This paper investigates dissociated sets, establishing bounds on their size within initial segments, analyzing the greedy algorithm's behavior, and exploring generalizations, thereby advancing understanding of their structure and construction.

Contribution

It provides new bounds on dissociated sets, proves the greedy algorithm's eventual doubling, and extends results to generalized dissociated sets.

Findings

Bounds on the size of dissociated sets within initial segments.

The greedy algorithm eventually doubles for constructing dissociated sets.

Generalizations of dissociated sets exhibit similar structural properties.

Abstract

A set $\mathcal S\subset \mathbb N$ is said to be a subset-sum-distinct or dissociated if all of its finite subsets have different sums. Alternately, an equivalent classification is if any equality of the form $$\sum_{s\in \mathcal S} \varepsilon_s \cdot s =0$$ where $\varepsilon_s \in \{-1,0,+1\}$ implies that all the $\varepsilon_s$'s are $0$. For a dissociated set $\mathcal S$, we prove that for $c_\ast = \frac 12 \log_2 \left(\frac \pi 2\right)$ and any $c_\ast-1<C<c_\ast$, we have $$\mathcal S(n) \,:=\, \mathcal S\cap [1,n] \,\le\, \log_2 n +\frac 12 \log_2\log_2 n + C$$ for all $n\in \mathcal N_C$ with asymptotic density $\mathbf d\left(\mathcal N_C\right)=2-2^{c_\ast-C}$. Further, we consider the greedy algorithm for generating these sets and prove that this algorithm always eventually doubles. Finally, we also consider some generalizations of dissociated sets and prove similar results about them.