The Quiet Geometry Hidden in Noise and Porosity
Highlights A new way to think about where curves live inside space; coarse tangent fields tie together large-scale structure with small-scale detail; dimension bounds link geometry to Nagata and Assouad dimensions; porous planar sets acquire simple tangent directions; a counterexample maps the limits of L2-type estimates. This is a collaborative work by Ball State University mathematician Guy C. David, Sylvester Eriksson-Bique of the University of Jyväskylä, and Raanan Schul of Stony Brook University, extending a lineage of ideas about tangents, traveling salesman geometry, and the shape of space itself.
1. A new way of seeing tangents in a wild universe
Gleaming lines appear whenever we zoom in on a smooth curve. If you take a smooth submanifold, its tangent space at a point captures all possible directions a tiny journey could begin. But what if the ambient set is not a neat curve or surface, but something far rougher — a porous planar set, a fractal-like dust of points, or a sprawling subset of a Hilbert space that might live in infinitely many dimensions? The question becomes trickier: can we still attach a sensible, directional guide to every point of such a set that makes sense across scales?
The paper from Guy C. David, Sylvester Eriksson-Bique, and Raanan Schul tackles this head-on by generalizing a line of thought that began with Alberti, Cs”ornyei, and Preiss. Their aim is not to force a perfect tangent plane everywhere, but to build two kinds of tangent notions that work in a broad, robust way: a coarse, scale-aware tangent field that works across many scales, and a pointwise, “weak” tangent field that mirrors the classical notion where it still makes sense. The setting is intentionally broad: we begin with subsets of Hilbert space (including infinite dimensions) and only assume the doubling property — a form of controlled local growth that stops geometric chaos from running away.
The authors’ team is a three-way collaboration: Guy C. David (Ball State University), Sylvester Eriksson-Bique (University of Jyväskylä), and Raanan Schul (Stony Brook University). Their work builds on the Analyst’s Traveling Salesman Theorem and the geometry of tangents to push beyond the plane and into the wildlands of higher dimensions.
2. Coarse tangent fields: a quantitative compass for space
Think of F as a multiresolution family: a nested zoo of balls that cover the set at all scales, like layers in a digital ore that reveal different features as you zoom in or out. The authors ask a precise, geometric question: can we assign to each ball a small, controlled linear subspace — a plane of dimension k ≤ d − 1 — so that, no matter which rectifiable curve Γ you pick, most of its interaction with the balls looks like it should align with the pre-assigned direction?
To formalize this, they introduce a coarse k-dimensional plane field, denoted τ, which assigns to each ball B a k-dimensional linear subspace τ(B). The twist is that when you compute how closely Γ sits near a line, you are allowed to restrict yourself to lines that are parallel to τ(B). This yields a coarse tangent field, defined with an inflation parameter A that scales the balls to AB(B). The field is said to be a coarse ε-tangent field if, roughly speaking, whenever Γ traverses the space, the sum of how far Γ strays from the closest line (in the restricted, τ-parallel sense) is controlled by the length of Γ, up to a factor that depends on ε and A.
What matters, and what makes the result powerful, is the dimension bound: if E badly fits d-planes (a way of saying E cannot be squeezed close to any d-dimensional flat object at any scale), then there exists a coarse ε-tangent field τ with dimension d − 1. In plain terms, you can fit a field of tangent directions that is just one dimension less than the obdurate dimension a set tries to hide behind, and you can do it with uniform control across scales. The proof leans on a Hilbert-space version of Jones’s traveling salesman machinery, a multiscale decomposition, and a careful analysis of how lines and planes can approximate curved objects inside a noisy space.
This coarse perspective is not just a mathematical gadget; it is a way to speak about structure without demanding perfection at every point. If the set is not too wild — if it doubles nicely and its Nagata or Assouad dimension is constrained — the tangent field becomes not merely possible but quantitatively controlled. In the plane, for porous sets like the Sierpiński carpet, the authors show one-dimensional coarse tangent fields can be chosen uniformly in ε. The punchline is that you can predict a single direction that captures the way curves weave through the set, at almost every scale, despite the set’s fractal-like gaps and holes.
The authors also show that their coarse theory is not a mere one-trick pony. They prove that the existence of a coarse tangent field with the right dimension is essentially tied to the property of badly fitting d-planes. If you can admit such a tangent field, E badly fits d-planes; conversely, if E badly fits d-planes, you can construct the coarse tangent field. This tight interplay reveals a deep bridge between dimension theory and coarse geometric approximation.
2.1 How coarse tangent fields are built (a glimpse behind the scenes)
The construction rides on a scaffold of dyadic cubes in metric spaces, due to Christ, which partitions space into nested cubes with well-behaved interfaces. The authors define a ball Q with center and radius, and they attach to it a ball B(Q) in the ambient Hilbert space. Then they choose a candidate plane τ(Q) of dimension up to d − 1, and they certify that, in a coarse sense, lines parallel to τ(Q) provide good approximations to Γ inside AB(Q). The core estimate is a summation over scales of how well the line approximates the curve, a Hilbert-space echo of Jones’s β-numbers, but now augmented to work in a coarse setting where the plane can vary with the location and scale. The argument threads a path through a multiscale decomposition, coronizations (a technique to peel away the parts of a set where geometry is too wild), and an intricate tracking of how planes interact with the curve’s geometry across scales.
One striking feature is the explicit link to Nagata and Assouad dimensions. If the underlying set has these dimensions strictly less than d, then it badly fits d-planes; this is the key lever that unlocks coarse tangent fields with dimension bounds independent of ε. In practical terms, it means that relatively tame dimensionality ensures we can pin down a finite, predictable family of tangent directions that govern the whole set’s geometry at all scales.
3. Pointwise weak tangent fields: a refined whisper of the tangent story
Beyond the coarse, the paper returns to the classical, pointwise notion of a tangent field in the plane: a line field along a set E that, for almost every curve Γ, matches the curve’s tangent directions. Alberti, Cs”ornyei, and Preiss showed that null sets in the plane admit a pointwise weak tangent field, a remarkable structural statement about how curves can pass through a “thin” set. David, Eriksson-Bique, and Schul push this line forward in a new direction: they prove that if E in a Hilbert space is a countable union of pieces that each badly fits d-planes, then E admits a d-dimensional pointwise weak tangent field. This is a robust, high-dimensional generalization of the idea that small, porous, or fractal pieces still carry a coherent tangent direction almost everywhere along curves that visit them.
Crucially, this is a structured, not-too-strong guarantee. It applies to countable unions of badly-fitting pieces, which covers many porous or fractal-like constructions. It does not claim that every high-dimensional null set admits an (n − 1)-dimensional pointwise weak tangent field, but it does offer a meaningful partial substitute in cases where a full planar analogue would be out of reach. The upshot is a toolkit for saying: even when sets are messy, there is a directional geometry left to exploit when we look at them through the lens of curves and their tangents.
4. Why this matters: from traveling salesman to data and beyond
The paper sits in a lineage that includes Jones’s traveling salesman theorem, which links a curve’s length to how well it can be approximated by a straight line across scales. The authors extend this philosophy to Hilbert spaces and to a coarse, quantitative language. The core result, in a nutshell, is: for every doubling subset E of a (finite or infinite dimensional) Hilbert space, you can construct a coarse tangent field whose dimension is governed by how E fits d-planes, with the approximation controlled through a multiresolution analysis. This is a generalization of the traveling salesman idea from curves in the plane to kaleidoscopic geometry in higher and infinite dimensions, carried out with a careful balance of scale, dimension theory, and measure-like quantities that track how a curve interacts with the ambient space.
But why should someone outside pure math care? Because geometry matters when you try to understand data. In many real-world datasets, the shape of the data manifold is not a smooth surface but a labyrinth drawn from real measurements, sometimes fractal-like, sometimes porous. If we want to compress, classify, or reason about such data, having a coarse tangent field is a kind of skip-list for geometry: a finite, scalable set of directions that captures the dominant orientation of paths through the data at many scales. It gives a lens to approximate Lipschitz mappings, and it fuels ideas about coarse differentials and 1-forms that tie geometry to functions in a way familiar to analysts but usable in higher dimensions and in more complex spaces.
One of the most striking technical moments is the development of coarse differentials of Lipschitz mappings. The authors outline how a coarse 1-form df, which pairs a tangent direction with a real value, can be defined and controlled by the coarse tangent field. This is not just an esoteric construction; it paves the way for a calculus on messy spaces where a traditional differential is not available. In a world full of noisy, high-dimensional data, coarse differentials could become a practical tool for understanding how a function changes along dominant directions across scales, a natural ally to multiscale data analysis and geometric learning.
There are also boundary notes that matter for theory and practice. The authors prove a sharp negative result: you cannot hope for a global L2-type bound in the strongest sense. They construct a compact, porous plane in the plane for which any coarse line field can fail to yield a finite L2 sum over all curves. This counterexample matters because it clarifies the limits of a certain kind of “normed” control over geometry. It cautions against over-optimistic generalizations and helps map the precise frontier where coarse tangent fields can be harnessed for quantitative estimates.
5. A broader horizon: implications for analysis, geometry, and computation
Reading this work, one is reminded of the interplay between structure and randomness that underpins much of modern analysis. The coarse tangent field concept is, in a sense, a pragmatic tool for taming the wildness that often appears when you step away from smooth manifolds into rough data or infinite-dimensional spaces. It aligns with broader themes in Lipschitz differentiability spaces, the study of how calculus can be done without the classical smooth structure, and the desire to understand how a map can be nearly linear along most curves inside a complicated set.
There is a particularly appealing bridge to computation. The construction relies on a hierarchical decomposition of space into dyadic cubes and a careful accounting of how curves interact with these cubes. This is not unlike the machinery behind wavelets, multiscale graphs, or hierarchical clustering, where the goal is to summarize complex geometry with a compact, scale-aware set of directions. If one were to translate these ideas into algorithms, there could be new ways to think about dimensionality, curvature, and local linearity in high-dimensional data, with potential applications in manifold learning, dimensionality reduction, or robust feature extraction for fractal-like data.
Beyond data science, the results touch deep questions about dimension theory itself. The Nagata and Assouad dimensions, two ways of quantifying how space can be covered and stitched together, sit at the heart of when and how a coarse tangent field can exist with a controlled dimension. This ties geometric measure theory to the structure of spaces that might eventually be used to model real-world phenomena, from porous materials to networks to high-dimensional function spaces in physics and engineering.
6. The human arc: where the math lives in institutions and minds
The work is a joint effort across continents and institutions. It reflects how modern mathematical exploration often travels through collaboration and cross-pollination: Ball State University’s Guy C. David, Finland’s Sylvester Eriksson-Bique at the University of Jyväskylä, and Stony Brook University’s Raanan Schul join forces to push the boundaries of tangent geometry in Hilbert spaces and beyond. The mathematical ideas are abstract, but the human arc is concrete: a group of researchers chasing a notion of tangency that survives messy reality and reveals a coherent, scalable structure in spaces large and small alike.
7. Looking forward: questions that still spark curiosity
No mathematical theory is complete, and this one ends with as many open doors as it opens. The authors pose natural follow-ups, such as whether the coarse tangent field can be chosen independently of ε in higher dimensions, or whether stronger summability properties can be attained in more delicate settings. They also wonder about the reach of these ideas in other Banach spaces or even metric spaces, and about how far the plane’s classical intuition can be translated to high-dimensional, or even infinite-dimensional, worlds without losing essential control.
And there are practical questions too. How might one computationally construct these coarse tangent fields in real data? Could coarse differentials of Lipschitz mappings become a standard tool in analyzing how functions behave along dominant directions across scales? The paper provides a rigorous foundation for exploring these possibilities, while also reminding us that geometry is full of surprises: there are limits, revealed by subtle counterexamples, to how neatly we can package high-dimensional geometry into tidy, universal estimates.
8. A final thought: math that teaches us to listen to structure
What makes this work compelling is less its final theorems than the way it reframes a familiar mathematical question — what does a tangent look like to a set that isn’t a smooth surface? — into a multi-layered story about coarse approximations, dimension, and scale. It tells us that even when a set resists a perfect linearization, there is a kind of honest, directional rhythm hiding in the noise, accessible through a carefully crafted, scale-aware field. It’s a reminder that geometry, in its deepest sense, is about finding the right perspective. When you broaden your view from a single tangent line to a whole coarse field that adapts with scale, you gain a powerful compass for navigating the infinite complexity of space. And that compass, in turn, could steer not just pure mathematics but the way we understand, compress, and analyze the patterns that shape our data-rich world.
In short, this collaboration—rooted in Ball State University, the University of Jyväskylä, and Stony Brook University—gives us a new way to sense the hidden geometry of space: a coarse map of directions that works across scales, binding together the wildness of porous sets, the rigor of dimension theory, and the practical dreams of multiscale analysis. It’s geometry at the edge of chaos, and it quietly asks us to listen a little more closely to the way space itself speaks through the curves that travel through it.
