<p>Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$A\subseteq [N]$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>A</mi> <mo>⊆</mo> <mo stretchy="false">[</mo> <mi>N</mi> <mo stretchy="false">]</mo> </mrow> </math></EquationSource> </InlineEquation> be such that for any pair of distinct subsets <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$B,C\subseteq A$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>B</mi> <mo>,</mo> <mi>C</mi> <mo>⊆</mo> <mi>A</mi> </mrow> </math></EquationSource> </InlineEquation>, the products <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\prod_{b\in B}b$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mo>∏</mo> <mrow> <mi>b</mi> <mo>∈</mo> <mi>B</mi> </mrow> </msub> <mi>b</mi> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\prod_{c\in C}c$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mo>∏</mo> <mrow> <mi>c</mi> <mo>∈</mo> <mi>C</mi> </mrow> </msub> <mi>c</mi> </mrow> </math></EquationSource> </InlineEquation> are distinct. We prove that <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$|A|\leq \pi(N)+\pi(N^{1/2})+o(\pi(N^{1/2}))$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow> <mo stretchy="false">|</mo> <mi>A</mi> <mo stretchy="false">|</mo> </mrow> <mo>≤</mo> <mi>π</mi> <mrow> <mo stretchy="false">(</mo> <mi>N</mi> <mo stretchy="false">)</mo> </mrow> <mo>+</mo> <mi>π</mi> <mrow> <mo stretchy="false">(</mo> <msup> <mi>N</mi> <mrow> <mn>1</mn> <mo stretchy="false">/</mo> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> <mo>+</mo> <mi>o</mi> <mrow> <mo stretchy="false">(</mo> <mi>π</mi> <mrow> <mo stretchy="false">(</mo> <msup> <mi>N</mi> <mrow> <mn>1</mn> <mo stretchy="false">/</mo> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\pi$</EquationSource> <EquationSource Format="MATHML"><math> <mi>π</mi> </math></EquationSource> </InlineEquation> is the prime counting function, answering a question of Erdős.</p>

Sharp bounds for sets with distinct subset products

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2025年12月25日

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R. Raghavan

摘要

Let $A\subseteq [N]$ $A \subseteq [N]$ be such that for any pair of distinct subsets $B,C\subseteq A$ $B, C \subseteq A$ , the products $\prod_{b\in B}b$ $\prod_{b \in B} b$ and $\prod_{c\in C}c$ $\prod_{c \in C} c$ are distinct. We prove that $|A|\leq \pi(N)+\pi(N^{1/2})+o(\pi(N^{1/2}))$ $| A | \leq π (N) + π (N^{1 / 2}) + o (π (N^{1 / 2}))$ , where $\pi$ $π$ is the prime counting function, answering a question of Erdős.