A trivariate real analytic function either has a rigid additive form or it strictly expands Hausdorff dimension.
Given three Borel sets in the real line with Hausdorff dimensions in [1/2, 1], take all triples (one point from each set) and apply a function f(x, y, z). How large is the image? For addition, f(x,y,z) = x+y+z, dimension sums — but structured functions can do less. The question: which functions guarantee dimension expansion, and which can keep the image small?
Pham (arXiv:2603.03567) proves a dichotomy. Either the function f has a specific rigid form — it decomposes as a function of additive combinations of the variables — or the image has strictly larger Hausdorff dimension than any of the input sets. There is no middle ground. Functions that are not additively decomposable are forced, by the geometry of their level sets, to spread the image across a larger-dimensional set.
The proof mechanism connects the geometry of the function's image to the regularity theory of Fourier integral operators. The dimension gain is not a counting argument but an analytic one: the curvature of the function's graph prevents the image from collapsing, and the optimal regularity bounds for the associated oscillatory integrals translate directly into dimension lower bounds.
The structural point: dimension expansion is generic. Most functions expand dimension. The functions that don't are exceptional — they have a specific algebraic structure (additive decomposition) that allows the image to remain thin. The theorem says: either you're in the exceptional class, or you strictly expand. The expansion is not gradual; it's a structural gap.
Pham, "On Hausdorff dimensions of k-point configuration sets and Elekes-Ronyai type theorems," arXiv:2603.03567 (2026).