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<p>In this paper we propose a dual version of the Furstenberg set problem and obtain partial results via <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(L^p\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mi>p</mi> </msup> </math></EquationSource> </InlineEquation> estimates of orthogonal projections. Examples are also discussed. Moreover, compared with general sets, we find that special structure like Cartesian product has better <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(L^p\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mi>p</mi> </msup> </math></EquationSource> </InlineEquation>-behavior. This leads to an improvement on some discretized sum-product estimates.</p>

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Orthogonal Projections, a Dual Furstenberg Set Problem, and Discretized Sum-Product Estimates

  • 期刊论文

  • 发表日期：

    2026年6月30日

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  • Longhui Li,
  • Bochen Liu

摘要

In this paper we propose a dual version of the Furstenberg set problem and obtain partial results via \(L^p\) L p estimates of orthogonal projections. Examples are also discussed. Moreover, compared with general sets, we find that special structure like Cartesian product has better \(L^p\) L p -behavior. This leads to an improvement on some discretized sum-product estimates.