Kakeya–Besicovitch sets in Lie groups and Kakeya sets in finite groups: the Murphy–Pakianathan questions

Murphy and Pakianathan (Topology and its Applications 180 (2015)) posed two group-theoretic analogues of the Kakeya problem. The Lie variant asks whether every Kakeya–Besicovitch subset of a connected Lie group $G$ — a Borel set containing a left translate of the unit segment of every one-parameter subgroup, with no continuity requirement on the translates — must have Hausdorff dimension equal to $\dim G$. The discrete variant asks for a sharp explicit constant $c(e,g)$ such that every Kakeya subset (a set containing a left coset of every cyclic subgroup) of every finite $g$-generated group of exponent dividing $e$ has size at least $c(e,g)|G|$, where the restricted Burnside theorem guarantees only that some positive constant exists. The universal form of the Lie conjecture is now refuted: Jing and Wu (2026) constructed algebraic Kakeya–Besicovitch sets of Hausdorff dimension $(d+r)/2 < d$ in every non-abelian compact semisimple Lie group of dimension $d$ and rank $r$. The surviving open core is precise and nonempty: full dimension remains undetermined for every non-compact Lie group (including the affine group, the smallest non-abelian case flagged by the source, and $\mathbb{R}^n$ for $n\ge 4$, where the question is exactly the classical Kakeya set conjecture); the optimal dimension in compact semisimple groups is open for rank $\ge 3$; and the discrete constant is open beyond the abelian special cases.

Background

A Kakeya set in $\mathbb{R}^n$ is a set containing a unit line segment in every direction. Besicovitch showed in 1928 that such sets can have Lebesgue measure zero, and the modern Kakeya set conjecture asserts that they must nevertheless be maximally large in the sense of fractal dimension: every Kakeya set in $\mathbb{R}^n$ has Hausdorff (and Minkowski) dimension $n$. Hausdorff dimension measures a set's size through optimal coverings by balls of vanishing radius; Minkowski dimension measures the growth rate of the number of $\delta$-balls needed to cover the set, and is always at least the Hausdorff dimension. The conjecture is known in dimensions 1 and 2 (Davies) and was settled in dimension 3 by Wang and Zahl (2025); it remains open for $n\ge 4$ and is intertwined with central open problems of Euclidean harmonic analysis such as the Fourier restriction conjecture.

Because affine lines are exactly the cosets of the one-dimensional subgroups of $(\mathbb{R}^n,+)$, Murphy and Pakianathan asked what survives of the Kakeya phenomenon when the ambient group is arbitrary. Let $G$ be a connected Lie group with Lie algebra $\mathfrak{g}$ (the tangent space at the identity); each $X\in\mathfrak{g}$ exponentiates to a one-parameter subgroup $\exp(tX)$, and directions are parametrized by the projective space $P(\mathfrak{g})$ of lines through the origin in $\mathfrak{g}$. Fix a positive-definite inner product on $\mathfrak{g}$ and the induced left-invariant Riemannian metric on $G$, and let $I(L)$ be the segment of length 1 centered at the origin along a line $L\in P(\mathfrak{g})$. A Kakeya–Besicovitch subset of $G$ is a Borel set $K$ such that for every $L\in P(\mathfrak{g})$ some left translate $g\cdot\exp(I(L))$ is contained in $K$; crucially, the translate may be chosen completely arbitrarily as a function of the direction — no continuity is required, exactly as in Besicovitch's classical construction. Murphy and Pakianathan note that the Hausdorff dimension of Borel subsets of $G$ is independent of the choice of left-invariant Riemannian metric, so the question is well posed. Their own theorems concern the continuous case, in which the translates vary continuously over the directions: in a connected nilpotent Lie group the underlying set of any continuous Kakeya line configuration must be all of $G$, and in a general connected Lie group any taut continuous configuration contains an open neighborhood of the identity, while any liftable (linear) continuous configuration contains the identity element. These rigidity results say nothing about the discontinuous Besicovitch-type sets, which is exactly what their question addresses. They single out the affine group of the line — the group of orientation-preserving affine transformations $x\mapsto ax+b$, $a>0$ — as the smallest non-abelian connected Lie group and the first case beyond the Euclidean one where an answer might be attainable.

For compact semisimple $G$ (a compact connected group with no nontrivial connected abelian normal subgroup, e.g. $SO(3)$ or $SU(n)$), write $d=\dim G$ and let $r$ be its rank, the dimension of a maximal torus (a maximal connected abelian subgroup). Jing and Wu (2026) determined that the natural global variant — sets containing a full left coset of every closed one-dimensional torus — has optimal Minkowski dimension exactly $(d+r)/2$. Their extremal set is $A_\theta=\{g\in G:\theta(g)=g^{-1}\}$, where $\theta$ is a Chevalley involution, an involutive automorphism of $G$ that inverts every element of a fixed maximal torus. Since $r<d$ for every non-abelian compact semisimple group, this exhibits genuine dimension compression, and the same set $A_\theta$ is a Kakeya–Besicovitch set of Hausdorff dimension $(d+r)/2$, refuting the naive full-dimension prediction for Kakeya–Besicovitch sets.

For the discrete variant, call a subset $E$ of a finite group $G$ a Kakeya set if it contains a left coset $gH$ of every cyclic subgroup $H\le G$. The restricted Burnside theorem of Zel'manov states that, up to isomorphism, there are only finitely many finite groups generated by $g$ elements whose exponent divides $e$ (i.e. with $x^e=1$ for every element $x$). Consequently some constant $c(e,g)>0$ exists with $|E|\ge c(e,g)|G|$ for every Kakeya set in every such group, but the theorem is purely qualitative — the constant it yields is the reciprocal of the size of the largest such group, which Murphy and Pakianathan call 'somewhat silly and definitely not sharp'. The source's request for a 'sharp, useful, explicit' formula leaves the optimality criterion open. The abelian prototype is Dvir's theorem, proved by the polynomial method — one interpolates a nonzero polynomial of low degree that vanishes on the complement of the set, and counting degrees then forces the set to be large: a Kakeya set in the vector space $(\mathbb{F}_q)^n$ (containing a line in every direction) contains at least a constant fraction of all $q^n$ points, with sharper constants due to Saraf and Sudan; Dummit and Hablicsek determined a sharp formula for products of two cyclic groups of order $p^k$, showing in particular that any valid constant must genuinely depend on the exponent $e$.

Problem Statement

The problem is the pair of questions posed by Murphy and Pakianathan, updated to the current state of the literature.

(1) Lie variant of the Kakeya conjecture. Let $G$ be a connected Lie group with Lie algebra $\mathfrak{g}$; fix a positive-definite inner product on $\mathfrak{g}$ and let $I(L)$ be the unit segment of a direction $L\in P(\mathfrak{g})$, the projective space of lines in $\mathfrak{g}$. A Kakeya–Besicovitch subset of $G$ is a Borel set $K\subseteq G$ such that for every $L\in P(\mathfrak{g})$ there exists $g\in G$ with $g\cdot\exp(I(L))\subseteq K$. The source asks: must $\dim_H K=\dim G$? The universal statement is now known to be false — Jing and Wu constructed algebraic Kakeya–Besicovitch sets of Hausdorff dimension $(d+r)/2 < d$ in every non-abelian compact semisimple Lie group — so the open content of the question is:

(a) For which connected Lie groups is full dimension still forced? Determine whether $\dim_H K=\dim G$ for every Kakeya–Besicovitch set $K$ in the affine group of the line, the group of orientation-preserving transformations $x\mapsto ax+b$ with $a>0$ — the smallest non-abelian connected Lie group and the case singled out by the source. The classification over all connected Lie groups, including the other non-compact (for example solvable) Lie groups, remains open. For simply connected abelian $G\cong\mathbb{R}^n$ this is exactly the classical Kakeya set conjecture, known for $n\le 3$ and open for $n\ge 4$.

(b) In a compact semisimple Lie group of dimension $d$ and rank $r\ge 3$, determine the exact optimal Hausdorff and Minkowski dimension of Kakeya–Besicovitch sets. The optimal Minkowski dimension is currently pinned only between the lower bounds $(d+1)/2$ (odd $r$) or $(d+2)/2$ (even $r$) and the construction-based upper bound $(d+r)/2$, which is conjectured sharp; for the Hausdorff dimension, only the same construction-based upper bound $(d+r)/2$ is currently known.

(2) Discrete variant. For a finite group $G$, call $E\subseteq G$ a Kakeya set if it contains a left coset of every cyclic subgroup of $G$. By the restricted Burnside theorem there are, up to isomorphism, only finitely many finite $g$-generated groups of exponent dividing $e$, so a constant $c(e,g)>0$ with $|E|\ge c(e,g)|G|$ for every such group and every Kakeya set $E$ in it exists; the constant obtained this way is well-defined but useless (the reciprocal of the largest group order in the class, a number the theorem guarantees to exist but does not effectively bound). Find a sharp explicit formula for $c(e,g)$ for finite $g$-generated groups of exponent dividing $e$.

Both parts pursue one scientific objective: quantify how much of a group a set must occupy when it contains a translate of every 'line' of the group — a unit one-parameter-subgroup arc in the Lie setting, a cyclic-subgroup coset in the finite setting — measured by Hausdorff/Minkowski dimension in the continuous setting and by normalized cardinality $|E|/|G|$ in the finite setting. Part (1) is open in the classes listed in (a) and (b); part (2) is open in general.

References

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