[Paper Review] A solution to a problem of Cassels and Diophantine properties of cubic numbers

Uri Shapira|ArXiv.org|Oct 23, 2008

Mathematical Dynamics and Fractals9 references4 citations

TL;DR

This paper resolves a long-standing problem posed by Cassels by proving that almost every pair of real numbers satisfies an inhomogeneous uniform version of Littlewood's conjecture, and further shows that pairs generating a totally real cubic number field also satisfy this property. The key result establishes that the inhomogeneous minimum of certain lattices is zero, implying dense orbit closures under diagonal actions, with applications to Diophantine approximation and homogeneous dynamics.

ABSTRACT

We prove that almost any pair of real numbers a,b, satisfies the following inhomogeneous uniform version of Littlewood's conjecture: (*) forall x,y in R, liminf_{|n| o\infty} |n| = 0, where denotes the distance from the nearest integer. The existence of even a single pair that satisfies (*), solves a problem of Cassels from the 50's. We then prove that if 1,a,b span a totally real number field, then a,b, satisfy (*). It is further shown that if 1,a,b, are linearly dependent over Q, a,b cannot satisfy (*). The results are then applied to give examples of irregular orbit closures of the diagonal groups of a new type.

Motivation & Objective

• To resolve Cassels' problem on whether the infimum of the spectrum of unimodular lattices in dimension $d \geq 3$ is zero.
• To establish a uniform inhomogeneous version of Littlewood's conjecture for almost all pairs $\alpha, \beta \in \mathbb{R}$, extending known results.
• To characterize when pairs $\alpha, \beta$ satisfying $1, \alpha, \beta$ linearly dependent over $\mathbb{Q}$ fail to satisfy the inhomogeneous condition.
• To demonstrate that certain algebraic pairs from totally real cubic fields satisfy the inhomogeneous condition, generalizing Cassels and Swinnerton-Dyer's result.
• To construct new examples of irregular orbit closures for diagonal groups in homogeneous spaces, using rigidity in higher-rank actions.

Proposed method

• Uses rigidity results for higher-rank commutative group actions on homogeneous spaces to analyze the behavior of lattice orbits.
• Applies the inhomogeneous minimum $\mu(x)$ of a lattice $x \in X_d$ as a measure of how close the lattice comes to the origin under the product norm $N(w) = \prod w_i$.
• Translates the Diophantine condition into a statement about the infimum of $|n| \langle n\alpha - \gamma \rangle \langle n\beta - \delta \rangle$ over $n \in \mathbb{Z}$, showing it vanishes for almost all $\alpha, \beta$.
• Employs the projection $\pi: Y_d \to X_d$ to relate grids (translates of lattices) to their underlying lattices, analyzing the product set $P(y)$ and its infimum $N(y)$.
• Applies Davenport's result on the lower bound of the inhomogeneous minimum in dimension 2 to construct counterexamples when $\dim_{\mathbb{Q}} \operatorname{span}\{1, v_1, \dots, v_{d-1}\} \leq 2$, showing $\mu(x_v) > 0$.
• Uses the fact that $\mu(x_0) \geq \mu(x)$ for $x_0 \in \overline{Ax}$ to rule out dense orbits when $\mu(x) > 0$, and constructs a counterexample to a conjecture on orbit closures in $X_3$.

Experimental results

Research questions

• RQ1 Does the inhomogeneous minimum $\mu(x)$ vanish for almost all lattices in $X_d$ when $d \geq 3$?
• RQ2 Can the inhomogeneous uniform version of Littlewood's conjecture be satisfied by almost all pairs $\alpha, \beta \in \mathbb{R}$, i.e., does $\liminf_{|n| \to \infty} |n| \langle n\alpha - \gamma \rangle \langle n\beta - \delta \rangle = 0$ hold for all $\gamma, \delta \in \mathbb{R}$?
• RQ3 Do pairs $\alpha, \beta$ such that $1, \alpha, \beta$ span a totally real cubic number field satisfy the inhomogeneous condition, even when $\gamma, \delta \neq 0$?
• RQ4 Under what algebraic conditions on $\alpha, \beta$ does the inhomogeneous condition fail, particularly when $1, \alpha, \beta$ are linearly dependent over $\mathbb{Q}$?
• RQ5 Can one construct examples of orbit closures for diagonal groups in $X_3$ that violate the conjecture that such orbits are either dense, closed, or contained in a closed orbit of an intermediate $\mathbb{Q}$-defined subgroup?

Key findings

• For $d \geq 3$, almost every lattice $x \in X_d$ satisfies $\mu(x) = 0$, affirmatively solving Cassels' problem on the spectrum being zero.
• For almost every $v \in \mathbb{R}^{d-1}$, the lattices $x_v$ and $z_v$ defined via unipotent matrices satisfy $\mu(x_v) = \mu(z_v) = 0$, implying the inhomogeneous condition holds.
• If $1, \alpha, \beta$ form a basis for a totally real cubic number field, then $\alpha, \beta$ satisfy the inhomogeneous uniform Littlewood condition: $\liminf_{|n| \to \infty} |n| \langle n\alpha - \gamma \rangle \langle n\beta - \delta \rangle = 0$ for all $\gamma, \delta \in \mathbb{R}$.
• If $\dim_{\mathbb{Q}} \operatorname{span}\{1, v_1, \dots, v_{d-1}\} \leq 2$, then $\mu(x_v) > 0$ and $\mu(z_v) > 0$, showing that the condition fails for algebraically dependent pairs.
• There exists $t \in \mathbb{R}$ such that the lattice $x_t \in X_3$ violates the conjecture that diagonal orbit closures are either dense, closed, or contained in a closed orbit of an intermediate $\mathbb{Q}$-defined subgroup.
• The orbit $Ax_t$ is neither dense nor closed, and is not contained in any closed orbit of an intermediate $\mathbb{Q}$-defined subgroup, providing a new type of irregular orbit closure.