number-theory / Extremal Number Theory
Let $A \subset \mathbb{N}$ be infinite with no distinct $a, b, c \in A$ such that $a \mid (b + c)$ with $b, c > a$. Can $|A \cap [1, N]|/\sqrt{N}$ have positive lower limit? Must every such $A$ fall below $N^{1-c}$ infinitely often?
Temporal state
No reconciled state yet.
Append-only history
parts (i) and (ii) resolved - a near-linear-density construction exists, refuting the N^{1-c} decay; the reciprocal-sum part remains open
Research memory
Let $A \subset \mathbb{N}$ be infinite with no distinct $a, b, c \in A$ such that $a \mid (b + c)$ with $b, c > a$. Can $|A \cap [1, N]|/\sqrt{N}$ have positive lower limit? Must every such $A$ fall below $N^{1-c}$ infinitely often?
parts (i) and (ii) resolved - a near-linear-density construction exists, refuting the N^{1-c} decay; the reciprocal-sum part remains open
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