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EP-1040: complete proof of the remaining vanishing question
I am submitting my paper, A solution to the Erdős Problem #1040, arXiv:2609.06050v1, submitted on 5 September 2026, as a complete proof of the remaining vanishing question in Erdős Problem #1040.
The paper establishes the capacity-one assertion for arbitrary compact subsets of the complex plane, removing the boundary regularity assumption from the earlier result. Together with the established capacity-greater-than-one case and the unbounded-set argument, it gives the affirmative vanishing conclusion for every closed infinite set of transfinite diameter at least one.
This completes the vanishing question in Erdős Problem #1040.
The paper credits the earlier results and perturbation strategy of Krishnapur, Lundberg, and Ramachandran, and the work of Ghosh and Ramachandran. The separate question of determination by capacity alone was already answered negatively by existing counterexamples, including those of Feng et al. and Ghosh and Ramachandran.
The work was developed through my prompting strategy using GPT-6-Astra, with AI assistance in mathematical reasoning and manuscript preparation.
Ioannis Tzachristas