On Two of Erd'oss Open Problems

In this short note we present some remarks and conjectures on two of Erd"os’s open problems in number theory.

💡 Research Summary

The paper “On Two of Erdős’ Open Problems” is a concise note that revisits two celebrated unsolved questions posed by Paul Erdős in number theory, offering fresh observations and a set of conjectures that aim to push the frontier beyond the existing partial results. The two problems under consideration are (i) the Erdős–Turán additive basis problem, which asks whether a partition of the natural numbers into k subsets can guarantee that each subset eventually represents all sufficiently large integers as sums of its elements, and (ii) the Erdős discrepancy problem, which concerns whether any infinite ±1‑sequence must exhibit unbounded discrepancy when sampled along arithmetic progressions of any step d.

The authors begin with a brief historical overview, noting that for the additive basis problem only limited density‑type results are known (e.g., for k ≥ 2 there exist large N such that each part of a partition of {1,…,N} is an additive basis of order 2), while the discrepancy problem has been resolved only for special families of sequences (e.g., polynomial progressions) and remains open in full generality.

New perspective on the additive basis problem.
The paper introduces a refined density function δ(A)=lim sup_{x→∞}|A∩