These are lecture notes for a mini-course given in Banff in June 2024. They discuss the problem of bounding the number of <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\delta $</EquationSource> <EquationSource Format="MATHML"><math> <mi>δ</mi> </math></EquationSource> </InlineEquation>-incidences <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\mathcal {I}_{\delta }(P,\mathcal {L}) := \{(p,\ell ) \in P \times \mathcal {L} : p \in [\ell ]_{\delta }\}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="script">I</mi> <mi>δ</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>P</mi> <mo>,</mo> <mi mathvariant="script">L</mi> <mo stretchy="false">)</mo> </mrow> <mo>:</mo> <mo>=</mo> <mrow> <mo stretchy="false">{</mo> <mrow> <mo stretchy="false">(</mo> <mi>p</mi> <mo>,</mo> <mi>ℓ</mi> <mo stretchy="false">)</mo> </mrow> <mo>∈</mo> <mi>P</mi> <mo>×</mo> <mi mathvariant="script">L</mi> <mo>:</mo> <mi>p</mi> <mo>∈</mo> <msub> <mrow> <mo stretchy="false">[</mo> <mi>ℓ</mi> <mo stretchy="false">]</mo> </mrow> <mi>δ</mi> </msub> <mo stretchy="false">}</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> under various hypotheses on <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$P \subset \mathbb {R}^{2}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>P</mi> <mo>⊂</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mn>2</mn> </msup> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\mathcal {L} \subset \mathcal {A}(2,1)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">L</mi> <mo>⊂</mo> <mi mathvariant="script">A</mi> <mo stretchy="false">(</mo> <mn>2</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. The main focus will be on hypotheses relevant for the Furstenberg set problem.

Approximate incidence geometry in the plane

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2025年9月5日

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Tuomas Orponen

摘要

These are lecture notes for a mini-course given in Banff in June 2024. They discuss the problem of bounding the number of $\delta $ $δ$ -incidences $\mathcal {I}_{\delta }(P,\mathcal {L}) := \{(p,\ell ) \in P \times \mathcal {L} : p \in [\ell ]_{\delta }\}$ $I_{δ} (P, L) : = {(p, ℓ) \in P \times L : p \in {[ℓ]}_{δ}}$ under various hypotheses on $P \subset \mathbb {R}^{2}$ $P \subset R^{2}$ and $\mathcal {L} \subset \mathcal {A}(2,1)$ $L \subset A (2, 1)$ . The main focus will be on hypotheses relevant for the Furstenberg set problem.