On Hausdorff dimensions of $k$-point configuration sets and Elekes-Rónyai type theorems
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866912941818773504 |
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| author | Pham, Minh-Quy |
| author_facet | Pham, Minh-Quy |
| contents | We prove a ''dimension expansion'' version of the Elekes-Rónyai theorem for trivariate real analytic functions: If $f$ is a trivariate real analytic function, then $f$ is either locally of the form $g(h(x)+k(y)+l(z))$, or the following is true: whenever a Borel set $A\subset\mathbb{R}$ has Hausdorff dimension $α\in \left(\frac{1}{2},1\right)$, $f(A\times A\times A)$ has dimension significantly larger than that of $A$, i.e. \begin{align*}
\dim_Hf(A\times A\times A)\geq α+\varepsilon(α),\quad \text{for some } \varepsilon(α)>0, \end{align*} Moreover, if $α>\frac{2}{3}$, $f(A\times A\times A)$ has positive Lebesgue measure. This is a considerable extension of the result established by Koh, T. Pham, and Shen (J. Funct. Anal. 286 (2024)).
We also obtain an alternative proof and an improvement for the Elekes-Rónyai type theorem for bivariate real analytic functions established by Raz and Zahl (Geom. Funct. Anal. 34 (2024)).
We derive these from more general results, showing that various $k$-point configuration sets of thin sets have positive Lebesgue measure by exploiting the optimal $L^2$-based Sobolev estimates for the associated family of Fourier integral operators. Extending the framework developed by Greenleaf, Iosevich, and Taylor (Mathematika 68 (2022), Math. Z. 306 (2024)) to prove Mattila-Sjölin type theorems, we obtain Falconer-type results for many configuration sets on which the method would be vacuous if demanding nonempty interior. In particular, when $k=2$, we generalize the Falconer-type result for metric functions in $\mathbb{R}^d$ satisfying strong non-vanishing curvature conditions established by Eswarathasan, Iosevich, and Taylor (Adv. Math. 228 (2011)) and the asymmetric Mattila-Sjölin type results of Greenleaf, Iosevich, and Taylor (J. Geom. Anal. 31 (2021)) to a broader class of smooth functions of asymmetric form. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_03567 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On Hausdorff dimensions of $k$-point configuration sets and Elekes-Rónyai type theorems Pham, Minh-Quy Classical Analysis and ODEs 28A75, 28A80, 58J40, 42B10 (primary), 52C10 (secondary) We prove a ''dimension expansion'' version of the Elekes-Rónyai theorem for trivariate real analytic functions: If $f$ is a trivariate real analytic function, then $f$ is either locally of the form $g(h(x)+k(y)+l(z))$, or the following is true: whenever a Borel set $A\subset\mathbb{R}$ has Hausdorff dimension $α\in \left(\frac{1}{2},1\right)$, $f(A\times A\times A)$ has dimension significantly larger than that of $A$, i.e. \begin{align*} \dim_Hf(A\times A\times A)\geq α+\varepsilon(α),\quad \text{for some } \varepsilon(α)>0, \end{align*} Moreover, if $α>\frac{2}{3}$, $f(A\times A\times A)$ has positive Lebesgue measure. This is a considerable extension of the result established by Koh, T. Pham, and Shen (J. Funct. Anal. 286 (2024)). We also obtain an alternative proof and an improvement for the Elekes-Rónyai type theorem for bivariate real analytic functions established by Raz and Zahl (Geom. Funct. Anal. 34 (2024)). We derive these from more general results, showing that various $k$-point configuration sets of thin sets have positive Lebesgue measure by exploiting the optimal $L^2$-based Sobolev estimates for the associated family of Fourier integral operators. Extending the framework developed by Greenleaf, Iosevich, and Taylor (Mathematika 68 (2022), Math. Z. 306 (2024)) to prove Mattila-Sjölin type theorems, we obtain Falconer-type results for many configuration sets on which the method would be vacuous if demanding nonempty interior. In particular, when $k=2$, we generalize the Falconer-type result for metric functions in $\mathbb{R}^d$ satisfying strong non-vanishing curvature conditions established by Eswarathasan, Iosevich, and Taylor (Adv. Math. 228 (2011)) and the asymmetric Mattila-Sjölin type results of Greenleaf, Iosevich, and Taylor (J. Geom. Anal. 31 (2021)) to a broader class of smooth functions of asymmetric form. |
| title | On Hausdorff dimensions of $k$-point configuration sets and Elekes-Rónyai type theorems |
| topic | Classical Analysis and ODEs 28A75, 28A80, 58J40, 42B10 (primary), 52C10 (secondary) |
| url | https://arxiv.org/abs/2603.03567 |