Summary

We prove Erdős’s conjecture: for any finite set with all subset sums distinct, the maximal element of must grow exponentially with the size of . More precisely, there exists a constant such that

Main Theorem

Theorem . Let be such that all subset sums are distinct. Then

Definitions

Let , , and . Define:

- Growth bound:

- Signed sum difference:

- Ramanujan sum:

- Correlation sum:

- Averaged correlation:

- Weighted average minimum:

where: - - -

Method

We study the function:

We use the identity:

We estimate the contribution over major arcs , proving:

This leads to the asymptotic bound:

Conclusion

This confirms Erdős’s conjecture by bounding from below by an explicit multiple of . The analytic method via Ramanujan sums and circle method opens new avenues in additive combinatorics.