Arithmetic structure of the exceptional set of projections
DOI:
https://doi.org/10.2140/mscand.2026.132.371Keywords:
projections, additive combinatorics, partial homogeneous Cantor setsAbstract
We study the arithmetic structure of the exceptional set of projections. For any bounded subset $E\subset \mathbb{R}^d$, let $$ \Omega=\{\xi\in \mathbb{R}: \dim_B(E+\xi E)=\dim_B E\}. $$ We prove that either $\Omega=\{0\}$ or $\Omega$ is a subfield of $\mathbb{R}$. We show that in general the statement does not hold for Hausdorff dimension and lower box dimension. Moreover, for any $s\in (0, 1]$ and a sequence $(r_k) \subset \mathbb{R}$, we construct an Ahlfors $s$-regular set $E\subset \mathbb{R}^2$ such that, for any $r_k, k\in \mathbb{N}$, we have \[ \overline{\dim}_B \{x+r_k y: (x, y)\in E\} <s. \]
References
J. Bourgain, “The discretized sum-product and projection theorems”, J. Anal. Math. 112 (2010), 193–236. https://doi.org/10.1007/s11854-010-0028-x
K. L. Chung, A course in probability theory, 3rd ed., Academic, San Diego, CA, 2001.
K. Falconer, Fractal geometry: mathematical foundations and applications, 2nd ed., Wiley, Hoboken, NJ, 2003. https://doi.org/10.1002/0470013850
K. Falconer, J. Fraser, and X. Jin, “Sixty years of fractal projections”, pp. 3–25 in Fractal geometry and stochastics V, edited by C. Bandt et al., Progr. Probab. 70, Springer, 2015. https://doi.org/10.1007/978-3-319-18660-3_1
D. Feng, Z. Wen, and J. Wu, “Some dimensional results for homogeneous Moran sets”, Sci. China Ser. A 40:5 (1997), 475–482. https://doi.org/10.1007/BF02896955
B. Green, “Sum-product phenomena in $F_p$: a brief introduction”, preprint, 2009. https://arxiv.org/abs/0904.2075
L. Guth, N. H. Katz, and J. Zahl, “On the discretized sum-product problem”, Int. Math. Res. Not. 2021:13 (2021), 9769–9785. https://doi.org/10.1093/imrn/rnz360
W. He, Sums products and projections of discretized sets, Ph.D. thesis, Université Paris-Saclay, 2017,. https://theses.fr/2017SACLS335
R. Kaufman, “On Hausdorff dimension of projections”, Mathematika 15 (1968), 153–155. https://doi.org/10.1112/S0025579300002503
J. M. Marstrand, “Some fundamental geometrical properties of plane sets of fractional dimensions”, Proc. London Math. Soc. (3) 4 (1954), 257–302. https://doi.org/10.1112/plms/s3-4.1.257
M. A. Martín and P. Mattila, “$k$-dimensional regularity classifications for $s$-fractals”, Trans. Amer. Math. Soc. 305:1 (1988), 293–315. https://doi.org/10.2307/2001054
P. Mattila, “Hausdorff dimension, orthogonal projections and intersections with planes”, Ann. Acad. Sci. Fenn. Ser. A I Math. 1:2 (1975), 227–244.
P. Mattila, Geometry of sets and measures in Euclidean spaces: fractals and rectifiability, Cambridge Studies in Advanced Mathematics 44, Cambridge Univ. Press, 1995. https://doi.org/10.1017/CBO9780511623813
P. Mattila, Fourier analysis and Hausdorff dimension, Cambridge Studies in Advanced Mathematics 150, Cambridge Univ. Press, 2015. https://doi.org/10.1017/CBO9781316227619
T. Orponen, “On the packing dimension and category of exceptional sets of orthogonal projections”, Ann. Mat. Pura Appl. (4) 194:3 (2015), 843–880. https://doi.org/10.1007/s10231-014-0401-y
T. Orponen, “On the projections of Ahlfors regular sets in the plane”, preprint, 2024. https://arxiv.org/abs/2410.06872
T. Orponen and P. Shmerkin, “On the Hausdorff dimension of Furstenberg sets and orthogonal projections in the plane”, Duke Math. J. 172:18 (2023), 3559–3632. https://doi.org/10.1215/00127094-2022-0103
T. Orponen and P. Shmerkin, “Projections, Furstenberg sets, and the ABC sum-product problem”, J. Amer. Math. Soc. 39:3 (2026), 857–913. https://doi.org/10.1090/jams/1073
K. Ren and H. Wang, “Furstenberg sets estimate in the plane”, preprint, 2023. https://arxiv.org/abs/2308.08819
P. Shmerkin and V. Suomala, Spatially independent martingales, intersections, and applications, Mem. Amer. Math. Soc. 1195, 2018. https://doi.org/10.1090/memo/1195
T. Tao and V. Vu, Additive combinatorics, Cambridge Studies in Advanced Mathematics 105, Cambridge Univ. Press, 2006. https://doi.org/10.1017/CBO9780511755149
M. Wu, “Projection theorems with countably many exceptions and applications to the exact overlaps conjecture”, preprint, 2025. https://arxiv.org/abs/2503.21923
Y. Zhao, Graph theory and additive combinatorics: exploring structure and randomness, Cambridge Univ. Press, 2023. https://doi.org/10.1049/cth2.12562
