Badly Approximable Matrices: Dynamics & Structures

• Badly approximable matrices are real m×n matrices that uniformly resist Diophantine approximation, characterized by a positive Lagrange constant.
• They are exceptional in Lebesgue measure yet achieve full Hausdorff dimension, linking metric number theory with fractal geometry.
• The topic extends to inhomogeneous affine forms, multiplicative analogues, and dynamical formulations, offering rich insights into approximation theory.

Searching arXiv for recent and foundational papers on badly approximable matrices and closely related multiplicative and inhomogeneous variants. arxiv_search.run({"^^^^^^^^3query3 approximable matrices\"^^^^^^^^3 OR ti:\3^^^^^^^^"badly approximable matrices\"","max_results":^^^^3all:\3query3^^^^,"sort_by":"submittedDate","sort_order":"descending"}) Found relevant arXiv results including recent work on badly approximable matrices, affine forms, multiplicative variants, and dimension theory. I will use the papers that overlap the provided corpus and cite them by arXiv id. Badly approximable matrices are real PRESERVED_PLACEHOLDER_^^^^3query3^^^^ matrices whose Diophantine approximation cannot be improved beyond the Dirichlet scale by any sequence of integer vectors. In the standard unweighted formulation, a matrix PRESERVED_PLACEHOLDER_^^^^3all:\3^^^^ is badly approximable if there exists PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3^^^^ such that

$\|Aq+p\|\ge c\,\|q\|^{-n/m}$

for all $p\in\mathbb Z^m$ and $q\in\mathbb Z^n\setminus\{0\}$; equivalently, with supremum norms,

$|q|^n\,|Aq-p|^m>c.$

The subject lies at the intersection of metric Diophantine approximation, homogeneous dynamics, transference theory, and fractal geometry, and now includes inhomogeneous affine forms, multiplicative analogues, higher-order notions defined via successive minima, and exact-order variants depending on a prescribed approximation function (&&&^^^^3query3^^^^&&&, &&&^^^^3all:\3^^^^&&&, &&&^^^^3 OR ti:\3^^^^&&&).

^^^^3all:\3^^^^. Classical formulation and structural properties

For $A\in M_{m,n}(\mathbb R)$, the classical badly approximable condition may be encoded by the Lagrange constant

$$\mathcal L(A)=\sup\Big\{\kappa>0:\ \|A\mathbf q-\mathbf p\|\,\|\mathbf q\| \ge \kappa \text{ for all } (\mathbf p,\mathbf q)\in\mathbb Z^m\times(\mathbb Z^n\setminus\{0\})\Big\},$$

and $A$ is badly approximable exactly when PRESERVED_PLACEHOLDER_^^^^3all:\3query3^^^^ (Simmons, 2015). In the same framework, singular matrices occupy the opposite extreme: PRESERVED_PLACEHOLDER_^^^^3all:\3all:\3^^^^ is singular if for every PRESERVED_PLACEHOLDER_^^^^3all:\3 OR ti:\3^^^^, eventually one can solve

PRESERVED_PLACEHOLDER_^^^^3all:\33^^^^

for all sufficiently large PRESERVED_PLACEHOLDER_^^^^3all:\3^^^^4 (&&&^^^^3query3^^^^&&&).

The classical theory distinguishes badly approximable matrices from merely Dirichlet improvable ones. The set PRESERVED_PLACEHOLDER_^^^^3all:\3^^^^5 consists of matrices with a uniform positive obstruction to improvement at the Dirichlet scale, while singular matrices satisfy arbitrarily strong improvements on all sufficiently large scales. Recent work on the Folklore set makes this distinction explicit by studying matrices that are Dirichlet improvable but neither badly approximable nor singular (Hussain et al., 2024).

A basic structural fact is transposition symmetry. A matrix is badly approximable if and only if its transpose is badly approximable, and the multiplicative analogue enjoys the same transpose invariance (German, 2010). This places badly approximable matrices within the transference tradition of Khintchine, Mahler, and Cassels, but with formulations adapted to both ordinary and multiplicative exponents.

^^^^3 OR ti:\3^^^^. Measure, Hausdorff dimension, and quantitative level sets

The global metrical picture has two complementary features. On the one hand, badly approximable matrices are exceptional in the Lebesgue sense; on the other hand, they are maximal in Hausdorff dimension. Schmidt’s theorem gives

PRESERVED_PLACEHOLDER_^^^^3all:\3^^^^6

and the asymptotic theorem of Fishman, Simmons, and Urbański refines this by studying the superlevel sets

PRESERVED_PLACEHOLDER_^^^^3all:\3^^^^7

For any norms on PRESERVED_PLACEHOLDER_^^^^3all:\3^^^^8 and PRESERVED_PLACEHOLDER_^^^^3all:\3^^^^9,

PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3query3^^^^

so

PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3all:\3^^^^

In particular, PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3 OR ti:\3^^^^ as PRESERVED_PLACEHOLDER_^^^^3 OR ti:\33^^^^, which recovers the Jarník–Schmidt theorem PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3^^^^4 (Simmons, 2015).

Earlier higher-dimensional estimates already showed that the codimension of fixed-constant level sets tends to PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3^^^^5 with the approximation constant, but without sharp asymptotics. Broderick and Kleinbock proved that for sufficiently small PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3^^^^6,

PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3^^^^7

with explicit PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3^^^^8 and constants PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3^^^^9 depending on $\|Aq+p\|\ge c\,\|q\|^{-n/m}$^^^^3query3^^^^ (&&&^^^^3all:\3^^^^&&&). The later asymptotic formula identifies the exact first-order coefficient in place of these unmatched upper and lower bounds (Simmons, 2015).

This metric picture is a common source of misconception. Full Hausdorff dimension does not imply positive Lebesgue measure, and the refined sets obtained by fixing an approximation constant are substantially thinner than the union over all constants. In that sense, the classical set $\|Aq+p\|\ge c\,\|q\|^{-n/m}$^^^^3all:\3^^^^ is simultaneously large and exceptional.

3. Inhomogeneous affine forms and badly approximable targets

The inhomogeneous theory fixes either a matrix $\|Aq+p\|\ge c\,\|q\|^{-n/m}$^^^^3 OR ti:\3^^^^ and varies the target $\|Aq+p\|\ge c\,\|q\|^{-n/m}$3, or fixes $\|Aq+p\|\ge c\,\|q\|^{-n/m}$4 and varies $\|Aq+p\|\ge c\,\|q\|^{-n/m}$5. For $\|Aq+p\|\ge c\,\|q\|^{-n/m}$6, one writes

$\|Aq+p\|\ge c\,\|q\|^{-n/m}$7

where $\|Aq+p\|\ge c\,\|q\|^{-n/m}$8 is $\|Aq+p\|\ge c\,\|q\|^{-n/m}$9-bad for $p\in\mathbb Z^m$^^^^3query3^^^^ if

$p\in\mathbb Z^m$^^^^3all:\3^^^^

Similarly,

$p\in\mathbb Z^m$^^^^3 OR ti:\3^^^^

is the exceptional set of matrices for a fixed target (&&&^^^^3all:\3query3^^^^&&&).

A central result is that these fixed-$p\in\mathbb Z^m$3 inhomogeneous sets are never full-dimensional in the ambient parameter space. For every $p\in\mathbb Z^m$4 and every $p\in\mathbb Z^m$5,

$p\in\mathbb Z^m$6

and in the doubly metric setting,

$p\in\mathbb Z^m$7

Moreover, if $p\in\mathbb Z^m$8, then $p\in\mathbb Z^m$9 is singular on average (&&&^^^^3all:\3query3^^^^&&&). Das, Fishman, and Simmons later made this picture quantitative: in the unweighted fixed-target setting,

$q\in\mathbb Z^n\setminus\{0\}$^^^^3query3^^^^

while for fixed $q\in\mathbb Z^n\setminus\{0\}$^^^^3all:\3^^^^ that is not singular on average,

$q\in\mathbb Z^n\setminus\{0\}$^^^^3 OR ti:\3^^^^

for all sufficiently small $q\in\mathbb Z^n\setminus\{0\}$3. They also proved the exact criterion that $q\in\mathbb Z^n\setminus\{0\}$4 has full Hausdorff dimension for some $q\in\mathbb Z^n\setminus\{0\}$5 if and only if $q\in\mathbb Z^n\setminus\{0\}$6 is singular on average (&&&^^^^3all:\3 OR ti:\3^^^^&&&).

The full inhomogeneous set $q\in\mathbb Z^n\setminus\{0\}$7 behaves differently from its fixed-$q\in\mathbb Z^n\setminus\{0\}$8 slices. It is well known that $q\in\mathbb Z^n\setminus\{0\}$9 is dense and has full Hausdorff dimension, and $|q|^n\,|Aq-p|^m>c.$^^^^3query3^^^^ is Schmidt-winning (&&&^^^^3all:\33^^^^&&&). Measure-theoretic behavior is subtler: if $|q|^n\,|Aq-p|^m>c.$^^^^3all:\3^^^^ is non-singular, then $|q|^n\,|Aq-p|^m>c.$^^^^3 OR ti:\3^^^^, but there are non-singular examples for which $|q|^n\,|Aq-p|^m>c.$3 has full measure with respect to some non-trivial algebraic measure on the torus. By contrast, under the stronger dynamical hypothesis that $|q|^n\,|Aq-p|^m>c.$4 has an accumulation sequence of length $|q|^n\,|Aq-p|^m>c.$5 and $|q|^n\,|Aq-p|^m>c.$6, one has

$|q|^n\,|Aq-p|^m>c.$7

on $|q|^n\,|Aq-p|^m>c.$8 (&&&^^^^3all:\33^^^^&&&).

4. Multiplicative theory and logarithmic thresholds

The multiplicative theory replaces the sup-norm control of the classical setting by a product condition. For $|q|^n\,|Aq-p|^m>c.$9, multiplicative bad approximability is defined by

$A\in M_{m,n}(\mathbb R)$^^^^3query3^^^^

where

$A\in M_{m,n}(\mathbb R)$^^^^3all:\3^^^^

The multiplicative transference theorem yields, in particular,

$A\in M_{m,n}(\mathbb R)$^^^^3 OR ti:\3^^^^

together with inequalities such as

$A\in M_{m,n}(\mathbb R)$3

for multiplicative Diophantine exponents (German, 2010).

A logarithmically sharpened version was developed by Fregoli. For $A\in M_{m,n}(\mathbb R)$4, let

$A\in M_{m,n}(\mathbb R)$5

where $A\in M_{m,n}(\mathbb R)$6. The threshold case is $A\in M_{m,n}(\mathbb R)$7. By Gallagher’s and Sprindžuk’s zero–one laws, $A\in M_{m,n}(\mathbb R)$8 has full Lebesgue measure for $A\in M_{m,n}(\mathbb R)$9 and measure $$\mathcal L(A)=\sup\Big\{\kappa>0:\ \|A\mathbf q-\mathbf p\|\,\|\mathbf q\| \ge \kappa \text{ for all } (\mathbf p,\mathbf q)\in\mathbb Z^m\times(\mathbb Z^n\setminus\{0\})\Big\},$$^^^^3query3^^^^ for $$\mathcal L(A)=\sup\Big\{\kappa>0:\ \|A\mathbf q-\mathbf p\|\,\|\mathbf q\| \ge \kappa \text{ for all } (\mathbf p,\mathbf q)\in\mathbb Z^m\times(\mathbb Z^n\setminus\{0\})\Big\},$$^^^^3all:\3^^^^. Fregoli proved that for every $$\mathcal L(A)=\sup\Big\{\kappa>0:\ \|A\mathbf q-\mathbf p\|\,\|\mathbf q\| \ge \kappa \text{ for all } (\mathbf p,\mathbf q)\in\mathbb Z^m\times(\mathbb Z^n\setminus\{0\})\Big\},$$^^^^3 OR ti:\3^^^^ with $$\mathcal L(A)=\sup\Big\{\kappa>0:\ \|A\mathbf q-\mathbf p\|\,\|\mathbf q\| \ge \kappa \text{ for all } (\mathbf p,\mathbf q)\in\mathbb Z^m\times(\mathbb Z^n\setminus\{0\})\Big\},$$3, and for every inhomogeneous shift $$\mathcal L(A)=\sup\Big\{\kappa>0:\ \|A\mathbf q-\mathbf p\|\,\|\mathbf q\| \ge \kappa \text{ for all } (\mathbf p,\mathbf q)\in\mathbb Z^m\times(\mathbb Z^n\setminus\{0\})\Big\},$$4,

$$\mathcal L(A)=\sup\Big\{\kappa>0:\ \|A\mathbf q-\mathbf p\|\,\|\mathbf q\| \ge \kappa \text{ for all } (\mathbf p,\mathbf q)\in\mathbb Z^m\times(\mathbb Z^n\setminus\{0\})\Big\},$$5

is everywhere dense in $$\mathcal L(A)=\sup\Big\{\kappa>0:\ \|A\mathbf q-\mathbf p\|\,\|\mathbf q\| \ge \kappa \text{ for all } (\mathbf p,\mathbf q)\in\mathbb Z^m\times(\mathbb Z^n\setminus\{0\})\Big\},$$6 and does not lie in any countable union of hyperplanes (&&&^^^^3 OR ti:\3^^^^&&&).

This theorem extends Moshchevitin’s earlier result for the two-dimensional case $$\mathcal L(A)=\sup\Big\{\kappa>0:\ \|A\mathbf q-\mathbf p\|\,\|\mathbf q\| \ge \kappa \text{ for all } (\mathbf p,\mathbf q)\in\mathbb Z^m\times(\mathbb Z^n\setminus\{0\})\Big\},$$7 to arbitrary dimensions. Its proof does not rely on Moshchevitin’s inductive method, since the needed higher-dimensional estimates for sums such as

$$\mathcal L(A)=\sup\Big\{\kappa>0:\ \|A\mathbf q-\mathbf p\|\,\|\mathbf q\| \ge \kappa \text{ for all } (\mathbf p,\mathbf q)\in\mathbb Z^m\times(\mathbb Z^n\setminus\{0\})\Big\},$$8

are not available in general. Instead, Fregoli develops a higher-dimensional Cantor-set construction inspired by Badziahin–Velani and uses an elementary geometric counting lemma for grid cubes intersecting hyperbolic sets (&&&^^^^3 OR ti:\3^^^^&&&).

The multiplicative threshold theorem also has an application to sums of reciprocals of fractional parts. For uncountably many matrices $$\mathcal L(A)=\sup\Big\{\kappa>0:\ \|A\mathbf q-\mathbf p\|\,\|\mathbf q\| \ge \kappa \text{ for all } (\mathbf p,\mathbf q)\in\mathbb Z^m\times(\mathbb Z^n\setminus\{0\})\Big\},$$9,

$A$^^^^3query3^^^^

which gives new higher-dimensional examples related to a question of Lê and Vaaler, though the bound is explicitly noted to be not optimal (&&&^^^^3 OR ti:\3^^^^&&&).

5. Dynamical formulations, successive minima, and higher order

The Dani correspondence identifies bad approximability with bounded diagonal-flow orbits on the space of unimodular lattices. Writing $A$^^^^3all:\3^^^^,

$A$^^^^3 OR ti:\3^^^^

one has

$A$3

where

$A$4

and $A$5 denotes the $A$6-th successive minimum (&&&^^^^3query3^^^^&&&). The same dynamical framework underlies the inhomogeneous theory, where one passes from lattices $A$7 to grids $A$8, and fixed-$A$9 bad approximation corresponds to orbits eventually staying in sets PRESERVED_PLACEHOLDER_^^^^3all:\3query3query3^^^^ of grids containing short nonzero vectors (&&&^^^^3all:\3query3^^^^&&&).

A higher-order hierarchy is obtained by replacing the first successive minimum with the PRESERVED_PLACEHOLDER_^^^^3all:\3query3all:\3^^^^-th. One defines

PRESERVED_PLACEHOLDER_^^^^3all:\3query3 OR ti:\3^^^^

These sets satisfy

PRESERVED_PLACEHOLDER_^^^^3all:\3query33^^^^

For every PRESERVED_PLACEHOLDER_^^^^3all:\3query3^^^^4, PRESERVED_PLACEHOLDER_^^^^3all:\3query3^^^^5 has Lebesgue measure PRESERVED_PLACEHOLDER_^^^^3all:\3query3^^^^6, while the successive gaps are full-dimensional: PRESERVED_PLACEHOLDER_^^^^3all:\3query3^^^^7 Thus the hierarchy is metrically thin at each level PRESERVED_PLACEHOLDER_^^^^3all:\3query3^^^^8, but each increment PRESERVED_PLACEHOLDER_^^^^3all:\3query3^^^^9 is as large as possible in both Hausdorff and packing dimension (&&&^^^^3query3^^^^&&&).

The modern dimension theory of these sets is closely tied to template methods and the variational principle of Das–Fishman–Simmons–Urbański. In this framework, successive minima functions are modeled by piecewise linear templates with slope constraints, and Hausdorff or packing dimension becomes an optimization problem over average contraction rates. This machinery appears both in the higher-order theory and in exact-order approximation problems for matrix sets defined by a function PRESERVED_PLACEHOLDER_^^^^3all:\3all:\3query3^^^^ (&&&^^^^3query3^^^^&&&, &&&^^^^3 OR ti:\33^^^^&&&).

6. Related variants, exact orders, and intermediate regimes

Several papers use the phrase “badly approximable matrices” in extended or paper-specific senses. In one line of work, for an approximation function PRESERVED_PLACEHOLDER_^^^^3all:\3all:\3all:\3^^^^ satisfying PRESERVED_PLACEHOLDER_^^^^3all:\3all:\3 OR ti:\3^^^^, one defines

PRESERVED_PLACEHOLDER_^^^^3all:\3all:\33^^^^

and

PRESERVED_PLACEHOLDER_^^^^3all:\3all:\3^^^^4

Here PRESERVED_PLACEHOLDER_^^^^3all:\3all:\3^^^^5 is the set of matrices that are approximable at scale PRESERVED_PLACEHOLDER_^^^^3all:\3all:\3^^^^6, but not at any strictly smaller scale PRESERVED_PLACEHOLDER_^^^^3all:\3all:\3^^^^7. Under monotonicity or condition PRESERVED_PLACEHOLDER_^^^^3all:\3all:\3^^^^8, if

PRESERVED_PLACEHOLDER_^^^^3all:\3all:\3^^^^9

then

PRESERVED_PLACEHOLDER_^^^^3all:\3 OR ti:\3query3^^^^

(&&&^^^^3 OR ti:\33^^^^&&&). This terminology is not the same as the classical set PRESERVED_PLACEHOLDER_^^^^3all:\3 OR ti:\3all:\3^^^^; it denotes an exact-order set.

A second refinement studies best approximation vectors PRESERVED_PLACEHOLDER_^^^^3all:\3 OR ti:\3 OR ti:\3^^^^, their norms PRESERVED_PLACEHOLDER_^^^^3all:\3 OR ti:\33^^^^, and remainder norms PRESERVED_PLACEHOLDER_^^^^3all:\3 OR ti:\3^^^^4. In the low-dimensional one-sided cases, bad approximability is characterized by bounded ratios: PRESERVED_PLACEHOLDER_^^^^3all:\3 OR ti:\3^^^^5 and

PRESERVED_PLACEHOLDER_^^^^3all:\3 OR ti:\3^^^^6

In higher dimensions, however, bounded ratios do not characterize bad approximability. There are matrices satisfying one bounded-ratio condition, or both, while failing bad approximability. For PRESERVED_PLACEHOLDER_^^^^3all:\3 OR ti:\3^^^^7 matrices, boundedness of both ratios still forces

PRESERVED_PLACEHOLDER_^^^^3all:\3 OR ti:\3^^^^8

and that bound is optimal (&&&^^^^3 OR ti:\3^^^^5&&&). This corrects a common extrapolation from the cases PRESERVED_PLACEHOLDER_^^^^3all:\3 OR ti:\3^^^^9 and PRESERVED_PLACEHOLDER_^^^^3all:\33query3^^^^.

The borderline between badly approximable, singular, and merely Dirichlet-improvable matrices has also become a distinct topic. The Folklore set

PRESERVED_PLACEHOLDER_^^^^3all:\33all:\3^^^^

is nonempty for all positive integers PRESERVED_PLACEHOLDER_^^^^3all:\33 OR ti:\3^^^^ except PRESERVED_PLACEHOLDER_^^^^3all:\333^^^^ and PRESERVED_PLACEHOLDER_^^^^3all:\33^^^^4, and in many regimes one can prescribe exact Dirichlet constants in a right neighborhood of PRESERVED_PLACEHOLDER_^^^^3all:\33^^^^5 (Hussain et al., 2024). This shows that bad approximability is not the only mechanism by which uniform improvement beyond Dirichlet can occur.

A further inhomogeneous characterization identifies classical bad approximability through approximation at every monotone divergent rate. In the matrix setting,

PRESERVED_PLACEHOLDER_^^^^3all:\33^^^^6

so a matrix PRESERVED_PLACEHOLDER_^^^^3all:\33^^^^7 is badly approximable exactly when, for every inhomogeneous parameter PRESERVED_PLACEHOLDER_^^^^3all:\33^^^^8, it cannot be inhomogeneously approximated at every monotone divergent rate (&&&^^^^3 OR ti:\3^^^^7&&&). This is a matrix-level extension of an earlier vector characterization (&&&^^^^3 OR ti:\3^^^^8&&&) and makes explicit the difference between homogeneous bad approximability and universal inhomogeneous approximability.

Taken together, these developments show that “badly approximable matrices” now denotes a core classical class together with a family of closely related threshold phenomena: fixed-constant slices, affine target fibers, multiplicative and logarithmic variants, higher-order successive-minima analogues, exact-order approximation sets, and intermediate Dirichlet-improvable regimes. The unifying theme is the persistence of a uniform obstruction to excessively good approximation, but the precise obstruction depends sensitively on whether one varies the target, the approximation function, the norm, the product structure, or the dynamical invariant being controlled.