Euclidean analogue for compact sets preserved by many special linear transformations

Prove that if a compact set E ⊂ R^2 has Hausdorff dimension α and is preserved by a set R ⊂ SL_2(R) of matrices with Hausdorff dimension greater than 3α/2, then E is contained in a line.

Background

The finite-field Theorem 1.3 shows that a subset of F_q2 with controlled size and a sufficiently large subgroup of SL_2(F_q) preserving it must have the rigid geometric structure of being contained in a line. The conjecture proposes an analogous statement in the Euclidean plane, replacing cardinality exponents by Hausdorff dimensions.

Specifically, the conjecture concerns a compact set E in R2 of Hausdorff dimension α and a set R of determinant-one linear transformations that preserve E pointwise as a set. The threshold dim_H(R) > 3α/2 is modeled on the finite-field exponent in Theorem 1.3.

References

In light of results on the packing set problems over the reals (see, for example, [3] and references therein) Theorem 1.3 suggests that it is reasonable to state the following conjecture. Conjecture 1.8. Let E be a compact set in R2 with dimH (E) = α. Assume that there exists a set R of matrices θ in SL2(R) such that θ(E) = E for each θ ∈ R, with dimH (R) > 3α/2. Then the set E is contained in a line. Here dimH means the Hausdorff dimension.

— Sets preserved by a large subgroup of the special linear group  (2501.01697 - Hung et al., 3 Jan 2025) in Conjecture 1.8, Section 1