Arya’s Algorithms

you can find the full paper here First Attempts, and the Limits of Energy When I first began thinking about angular occupancy, I wasn’t looking for a brand-new problem. I was chasing a theme that has long fascinated me in discrete geometry: how do we measure the geometric richness of a finite set of points? This fascination is not unique to me. Erdős, way back in 1946, posed the distinct distances problem : given n points in the plane, how many distinct distances can they determine? That question alone gave birth to entire decades of work in combinatorial geometry. Later, Fishburn and Füredi asked about distinct angles , while Pach and Sharir studied repeated angles . Each time, the same underlying itch was being scratched: when you place points in the plane, what kinds of diversity can they generate? Why angles, why bins? It was natural to think of angles — after all, distances and directions had already been well studied. But if you go straight for “distinct angles,” you run in...

  Measuring Angular Diversity: ε-Angle Classes in Planar Point Sets One of the joys of combinatorial geometry is how simple questions about points in the plane open doors to deep mathematics. From Erdős’s famous problem on distinct distances, to the countless studies of directions, incidences, and angles, each setting turns a finite set of points into a rich structure of combinatorial possibilities. This blog explores a new addition to that landscape: the ε-angular occupancy function , introduced in the paper “On ε-Angle Classes Determined by Planar Point Sets.” At heart, it asks: Given a set of points, how many “angle classes” do they determine if we only measure angles up to a resolution ε? This question blends discrete geometry with an “approximate” perspective: not every angle needs to be distinct, just distinguishable at scale ε. From Distances to Angles Let’s recall some classical milestones in the study of geometric diversity: Distinct distances problem (Erdős...

  Nowhere Differentiability at the Edge: The Critical Takagi Function When we think of a “pathological function” in analysis, the usual suspects come to mind: Weierstrass’s nowhere differentiable monster, Cantor’s dust of measure zero, or perhaps the Koch snowflake with its infinitely jagged boundary. Each shows us that functions and shapes can behave in ways our geometric intuition never expected. Among these classics lives the Takagi function , a beautifully simple series that hides fractal complexity. In recent years, mathematicians have generalized it, producing an entire family of functions whose regularity depends on two parameters. And the paper we’re discussing today — “Nowhere Differentiability of the Generalized Takagi Function at the Critical Threshold” — nails down the last missing piece of the puzzle: what happens exactly at the delicate “critical” parameter setting. The Takagi Function and Its Generalization The original Takagi function was introduced in 1903...

  My Journey into Constructions of Pathological Functions When I first learned about continuous but nowhere differentiable functions, I was fascinated. The idea that you could draw a function without lifting your pencil, yet never find a tangent line anywhere, felt like a paradox. How could such functions exist? And more importantly — could I discover one myself, using only the tools I knew? This question eventually led me to write my paper Functional Equations and Combinatorial Constructions of Pathological Functions in Real Analysis . But the road there was more like a chain of small discoveries than a single big idea.       check out the full paper The Starting Point: Weierstrass and Takagi My journey began with the Weierstrass function : W ( x ) = ∑ n = 0 ∞ a n cos ⁡ ( b n π x ) . W(x) = \sum_{n=0}^\infty a^n \cos(b^n \pi x). It’s elegant, but all the proofs I could find that W W is nowhere differentiable used harmonic analysis . Fourier series, lac...

  Functional Equations and Combinatorial Constructions of Pathological Functions in Real Analysis By Arya Dubey                                                                                                                                      Check out the full paper Abstract The existence of continuous nowhere differentiable (CND) functions remains one of the most striking and counterintuitive phenomena in real analysis. These objects are continuous everywhere but possess no derivative at any point. Once regarded as pathological monsters, they now arise naturally in probability, dynamics, fractal geometry, and harmonic analysis. This blog post ...