number-theory / Additive combinatorics
For every finite set $A\subset\mathbb Z$ with $|A|\ge 2$, define $$C(A)=\frac{\log\left(|A+A|/|A|\right)} {\log\left(|A-A|/|A|\right)}.$$ Determine the largest possible value of $C(A)$, equivalently the least universal exponent $c$ such that $$\frac{|A+A|}{|A|} \le \left(\frac{|A-A|}{|A|}\right)^c$$ for every such set $A$. The result proves that the supremum is exactly $2$, although no individual admissible set attains it.
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New arXiv preprint with an author-provided Lean formalization; not yet peer-reviewed.
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For every finite set $A\subset\mathbb Z$ with $|A|\ge 2$, define $$C(A)=\frac{\log\left(|A+A|/|A|\right)} {\log\left(|A-A|/|A|\right)}.$$ Determine the largest possible value of $C(A)$, equivalently the least universal exponent $c$ such that $$\frac{|A+A|}{|A|} \le \left(\frac{|A-A|}{|A|}\right)^c$$ for every such set $A$. The result proves that the supremum is exactly $2$, although no individual admissible set attains it.
New arXiv preprint with an author-provided Lean formalization; not yet peer-reviewed.
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