HBP | Base | 0.1 • Blur as a Universal Principle: Number Theory, Probability, Dynamics

Blur as a Universal Principle: Number Theory, Probability, Dynamics Aleksandar Perišić August 2025 Abstract We advocate a simple organizing principle: blur sharp or oscillatory objects by positive, normalized approximate identities (Poisson/Fejér/Gaussian/Abel), prove the smoothed statement by positivity and dominated (or mean–ergodic) convergence, and then unsmooth by a standard deconvolution/Tauberian step. This two–move template explains a surprising range of classical results. On the circle, Poisson blurring turns Weyl equidistribution of {nα} into a one–line argument via the decay of fixed Fourier modes and Pr|k|<∞ , with equidistribution recovered as r↑ 1. In probability, convolving characteristic functions with the Poisson kernel is equivalent to testing against x7→ e−y|x| ; Lindeberg replacement at fixed blur and an equi–Lipschitz bound yield the Gaussian limit by Lévy’s theorem. In ergodic theory, Abel averages are positive L1 contractions; Dunford–Schwartz gives convergence in norm, and Abel → Cesàro plus the Hopf maximal inequality recovers Birkhoff’s pointwise ergodic theorem. We also prove a blur/ ε – δ equivalence: classical limits are equivalent to convergence of Gaussian blurs together with vanishing local oscillation. Finally, defining a blur–integral R1 0 ( f∗ϕy )and letting y↓ 0reproduces the Lebesgue integral for every bounded measurable f and therefore extends the Riemann integral; Dirichlet’s function becomes integrable by blur. Beyond these worked examples we argue that blur is not just a technical trick but an epistemic necessity: whenever a theory learns from a world it does not fully control, there is an inescapable “band of ignorance” that must be handled honestly. We outline three structural places where this band appears—between addition and multiplication, between finite and infinite, and in the process of theorem discovery itself—and explain how these all reduce to a single small blur budget. Introduction “Blurring” means replacing sharp cutoffs or oscillatory kernels by positive, normalized approximate identities (such as the Poisson or Fejér kernel). This simple step turns difficult convergence problems into immediate consequences of positivity and dominated convergence. We illustrate the universality of this principle, its blurriversality, in five contexts: 1. Equidistribution of sequences in number theory. 2. The central limit theorem in probability. 3. The ergodic theorem in dynamics. 4. Standard ε–δlimits. 5. Riemann and Lebesgue on the Dirichlet example. In words: blur replaces sharp boundaries by soft ones; proving limits for the blurred objects is straightforward, and a standard “unsmoothing” step recovers the sharp theorem. Notation. {x} denotes the fractional part of x .1 A is the indicator of a set A . f∗g is convolution. Weak convergence of laws is written ⇒. 1 Setup Let α∈R\Q . Classical Weyl’s theorem states that {nα} is uniformly distributed modulo 1. Traditional proofs use exponential sums and delicate bounds. We instead apply a frequency blur on the circle. Blur kernel Fix 0< r < 1. Let Pr(θ)be the periodic Poisson kernel Pr(θ) = X k∈Z r|k|eikθ =1−r2 1−2rcos θ+r2≥0,1 2πZπ −π Pr(θ)dθ = 1. In words: Pr is a smooth bell on the circle with total mass 1; the parameter r damps high frequencies. For xn:= {nα}consider the blurred empirical density WN,r(θ) := 1 N N X n=1 Pr2π(xn−θ 2π)(θ∈R/2πZ). In words: WN,r is the smoothed histogram of the points xn, evaluated at angle θ. By the Fourier expansion of Pr, WN,r(θ) = X k∈Z r|k|SN(k)e−ikθ, SN(k) := 1 N N X n=1 e2πikxn=1 N N X n=1 e2πiknα. In words: the smoothed histogram is controlled by the exponential sums SN ( k ), with r|k| downweighting high-frequency modes. Analysis For k = 0 we have SN (0) = 1, hence the zeroth Fourier coefficient of WN,r is 1. For each fixed k= 0, SN(k) = e2πikα(1 −e2πikNα) N(1 −e2πikα)−−−−→ N→∞ 0, In words: the geometric-series formula shows every nonzero Fourier mode averages out. Therefore, for each fixed r < 1, WN,r(θ)uniformly in θ −−−−−−−−→ N→∞ 1. In words: at any fixed blur level, the smoothed histogram becomes flat; letting r↑ 1removes the blur and yields equidistribution. Remark 1. No Diophantine bounds on sharp exponential sums were needed: the identity gives SN ( k ) → 0for each k = 0, positivity forces WN,r → 1, and unsmoothing r↑ 1gives uniform distribution. Setup Let X1, X2, . . . be i.i.d. with EX1 = 0, Var ( X1 ) = 1. Classical CLT: Sn/√n⇒N (0 , 1) where Sn = Pn j=1 Xj . Traditionally, one studies characteristic functions φn ( t ) = E [ eitSn/√n ]and proves φn(t)→e−t2/2pointwise. We instead blur in t. 2 Blurred characteristic functions Fix y > 0and the (continuous) Poisson kernel Py ( t ) = 1 π y y2+t2 with Fourier transform c Py ( ξ ) = e−y|ξ|. Define Bn,y := ZR Py(t)φn(t)dt =Ehe−y|Sn/√n|i, by Fubini and Fourier inversion. In words: Bn,y tests the law of Sn/√n against the simple bounded function x7→ e−y|x|, i.e., a gently blurred probe of the bulk. Limit law from blur For each fixed y > 0, Bn,y −→ ZR e−y|x|e−x2/2 √2πdx (n→ ∞), In words: the blurred probes converge to the same probes of a standard normal—hence the limit is Gaussian at every blur scale. Unsmoothing The family (φn)nis equi-Lipschitz: |φ′ n(t)| ≤ Eh|Sn| √ni≤1. In words: if all blurred versions of the curves converge and the curves cannot wiggle too fast, then the curves themselves converge; Lévy’s theorem then gives the CLT. Remark 2. Blur kills high-frequency oscillations; only local variance data remain. One can also deduce the CLT via standard smoothing inequalities (Berry–Esseen) with gy. Setup Let (X, B, µ, T)be measure-preserving, and f∈L1(µ). Birkhoff’s theorem states that 1 N N−1 X n=0 f(Tnx)−→ E(f| I)(x)for a.e. x, where Iis the T-invariant σ-algebra; in particular, if Tis ergodic, the limit equals Rf dµ. The sharp cutoff 1 [0,N) is delicate. With blur (Abel averages), convergence is immediate at the level of linear-operator theory. Blurred averages Fix λ∈(0,1), and set exponentially blurred averages Aλf(x) := (1 −λ)∞ X n=0 λnf(Tnx). In words: this is a geometrically weighted moving average along the orbit—i.e., a time average with a gentle exponential blur. 3 Analysis By the Dunford–Schwartz mean ergodic theorem (for positive contractions), Aλf→E ( f| I )in L1as λ↑1. By a standard Abel–Cesàro Tauberian argument together with the Hopf maximal ergodic inequality, the Cesàro averages converge almost everywhere to the same limit. In words: prove convergence first for the blurred averages (easy), then remove the blur to recover the usual block averages—this is Birkhoff’s theorem. Remark 3. Positivity +normalization +approximate identity in time (Abel kernel) place us in the contractive Markov-operator setting where norm convergence to the invariant projection is automatic; the Tauberian step and the ergodic maximal inequality give pointwise convergence. Blur-based limits and their equivalence to ε–δ Let G ( u ) = π−1/2e−u2 and Gy ( u ) = y−1G ( u/y )so that Gy≥ 0, RRGy = 1, and Gy→δ0 (Dirac) as y↓0. For locally integrable fdefine the Gaussian blur at scale yby By[f](x0) := ZR f(x0+u)Gy(u)du. In words: By[f](x0)is the average of fnear x0with a Gaussian window of width y. Theorem 4 (Blur ⇐⇒ classical limit under minimal regularity).Fix x0∈R and let f be locally bounded near x0. (i) Classical ⇒ blur. If limx→x0f ( x ) = L , then limy↓0By [ f ]( x0 ) = L .In words: ordinary limits imply blurred limits. (ii) Blur +vanishing local oscillation ⇒ classical. If limy↓0By [ f ]( x0 ) = L and f does not oscillate arbitrarily fast near x0 , then limx→x0f ( x ) = L .In words: if all Gaussian averages settle and fis not wildly wiggly, then fitself settles to the same value. Proof. (i) Split the blur integral into |u|< δ and |u| ≥ δ ; boundedness and Gaussian tails give the claim by dominated convergence. (ii) Use a small neighborhood where f varies by at most ε , note that the Gaussian tail outside that neighborhood has tiny mass, and decompose f ( x ) −L into “value minus blur at x ” +“blur shift from xto x0”+“blur at x0minus L”. Remark 5 (Sharpness and scope).(1) Part (i) uses only local boundedness and vanishing Gaussian tails. (2) Some oscillation control is necessary; e.g. f ( x ) = sin (1 /x )at x0 = 0 has By→ 0but no classical limit. (3) Equivalently, convergence of By uniformly over tiny shifts and small local oscillation together characterize the classical limit. Remark 6 (Generality of blur).The Gaussian is only one convenient blur. Any family of normalized, positive kernels concentrating to a Dirac delta works (compactly supported mollifiers, Poisson kernels, or randomized smoothing). Blur reconciles Riemann and Lebesgue on the Dirichlet example Let f =1 Q∩[0,1] (Dirichlet’s function: f ( x )=1on rationals, 0on irrationals). Classically: f is not Riemann integrable on [0 , 1] (upper sums = 1, lower sums = 0), but the Lebesgue integral exists and equals 0since λ(Q)=0. 4 Blur kernel and blur-integral. Let ϕ∈L1 ( R )be nonnegative with Rϕ = 1, and set ϕy(u) = y−1ϕ(u/y)(y > 0). Extend fby 0outside [0,1], and define (f∗ϕy)(x) = ZR f(x−u)ϕy(u)du, x ∈R. In words: (f∗ϕy)(x)averages nearby values of faround xusing the window ϕy. Define the blur-integral Iblur(f) := lim y↓0Z1 0 (f∗ϕy)(x)dx. In words: integrate a smooth surrogate of fand then let the blur vanish. Theorem 7 (Blur = Lebesgue (and extends Riemann)).For every bounded measurable f on [0,1] (extended by 0outside), Iblur(f)exists and equals Z[0,1] f dλ. In words: blur-integration reproduces the Lebesgue integral and therefore extends the Riemann integral. Proof. Mass is preserved by convolution, and boundary leakage from (0 , 1) under ϕy vanishes as y↓0, so the integral of f∗ϕyover [0,1] tends to the Lebesgue integral of fon [0,1]. Remark 8 (What the blur did).For Dirichlet’s f =1 Q∩[0,1] , the raw Riemann procedure fails because f is discontinuous at every point: each interval contains both rationals and irrationals, so every upper sum is 1and every lower sum is 0no matter how fine the partition. In particular, no choice of mesh size ∥P∥makes the Riemann upper and lower sums agree. Blur replaces fby the mollified surrogate (f∗ϕy)(x) = ZR f(x−u)ϕy(u)du =Z[0,1] 1Q(t)ϕy(x−t)dt, where ϕy ( u ) = y−1ϕ ( u/y )is nonnegative with RRϕy = 1. For each fixed y > 0the map x7→ ( f∗ϕy )( x )is continuous on [0 , 1], hence Riemann integrable, and convolution preserves total mass: Z1 0 (f∗ϕy)(x)dx =ZR f(t)Z1 0 ϕy(x−t)dxdt −−→ y↓0Z[0,1] f(t)dλ(t). Since λ ( Q∩ [0 , 1]) = 0, the Lebesgue integral on the right is 0. In other words, blur smooths f into a continuous profile f∗ϕy that is classically integrable, keeps the total mass Rf unchanged, and then collapses back to the Lebesgue integral of f as y↓ 0. The pathology is not hidden; it is averaged away in a controlled manner that exactly matches the Lebesgue notion of “size”. It is worth reminding the reader that real numbers are an idealized completion of their rational origins: once we choose to observe an infinite continuous spectrum, purely discrete sets of measure zero become invisible to the integral, and conversely. The blur procedure makes that trade-off explicit—a small uncertainty principle between discrete and continuous descriptions. Remark 9 (Stochastic blur (randomized smoothing)).Let Uy have density ϕy and set By [ f ]( x ) = E [ f ( x + Uy )]. Then R1 0By [ f ]( x ) dx →R[0,1] f dλ ; Monte Carlo sampling of f ( x + Uy )approximates the same limit. Remark 10 (Edges and “returning to the domain”).One may also periodize f , use compactly supported ϕ , or adopt boundary renormalizations; all give the same limit because ϕy has unit mass and its leakage tends to zero. 5 Remark 11 (Information monotonicity (“no free lunch”)).For essentially bounded f and c > 0, letting τy(c)be the tail mass of the kernel outside |u| ≤ cy, ess inf |u|≤cy f(·+u)−2∥f∥∞τy(c)≤Byf≤ess sup |u|≤cy f(·+u)+2∥f∥∞τy(c). In words: a blur output lies within the local range of f (up to a vanishing tail), so blur cannot create new structure. Remark 12 (Preparation and fit).Choose kernels with nonnegative transform, operate where analytic decompositions are controlled, and pick y so signal dominates noise before sending y↓ 0. Blur beyond the worked examples: epistemic background The previous sections showed blur at work in concrete theorems: equidistribution, the CLT, ergodic limits, ε – δ convergence, and Lebesgue integration. But blur is not an occasional analytic trick; it is a structural response to the fact that we always learn under partial information. This section sketches that epistemic backdrop and three places where blur shows up before any particular theorem is written down. Blur as an inescapable ingredient of learning The world, for our purposes, is whatever produces the data we try to understand. A theory is a lens we put between ourselves and that world. Between “what the theory already knows” and “what we cannot hope to predict” there is typically a thin, structured band of ignorance that is still readable: some coarse trends survive, fine details do not. Blur is the conscious decision to work at that band: choose a kernel, fix a budget, and only trust quantities that are stable under that operation. In a separate essay on randomness, blur, and the □ -operator, this is made explicit: “true randomness” is treated as a limiting, zero-information ideal □ , and every real model we use is a blurred approximation to that ideal, carrying a few honest bits of structure. The role of blur is to name those bits and keep them visible. Why the same mathematics keeps reappearing Historically it is striking that very different cultures converged on essentially the same basic mathematical toolkit: integer arithmetic, Euclidean geometry, rudimentary limits, primes. These arose in environments with different religions, economies, and technologies, yet the core objects agree to a surprising extent. From the blur viewpoint this is less mysterious. Certain operations are singled out not by history but by the way they organize ignorance. Addition and multiplication are the obvious examples. They form two complementary channels—one additive, one multiplicative—that are both incredibly expressive and, as it turns out, cannot be simultaneously sharp. Addition, multiplication, and a built-in uncertainty principle In the soft-addition/soft-multiplication framework, addition lives naturally on ( R, dx )and multiplication on ((0 ,∞ ) , dx/x ). After the logarithmic change of variables, the Mellin transform becomes a Fourier transform in log x ; the usual Heisenberg principle then says that we cannot localize sharply in both channels at once. Switching between additive and multiplicative descriptions necessarily introduces a channel-switch error, a small blur in the log-variable that cannot be removed without paying a price elsewhere. What looks like an obstacle is also an opportunity. The unavoidable uncertainty between addition and multiplication means there is a permanently unexplored core where new structure 6 can live. This helps explain why the additive/multiplicative pair is so central across cultures: they are precisely the operations that both model a huge portion of the world and refuse to exhaust it. Blur here is not cosmetic; it is the toll we pay to use both channels honestly. Blur at the finite–infinite interface A second structural source of blur appears at the interface between finite and infinite. In practice we move freely between finite initial segments of N and the infinite whole: we remove one element at a time, we speak about “all large enough n ”, we swap sums and limits, and so on. Conceptually this involves switching lenses: looking at the same successor chain once as a finite object, once as an infinite one. In the note on blur at the finite–infinite interface this is formalized via Abel weights wq ( n ) = (1 −q ) qn . For 0 < q < 1they define a finitely additive probability µq on the successor orbit with the resolvent identity µq(E) = (1 −q)1E(0) + q µq(S−1E), and the property that finite sets have µq -mass → 0while cofinite sets have mass → 1as q↑ 1. In this setting, the mere existence of a Peano successor orbit is equivalent to the existence of such a blur. Informally: every time we treat “finite” and “cofinite” as complementary shapes that can be honestly compared, we are already paying for an Abel-type blur. Regularizations of series and integrals are one visible reminder of that price: we ignore some partial sums or extract a constant and accept that this is not a mirage but a structured blur at the finite–infinite seam. Blur in the process of discovery A third place where blur is hiding is in the way we discover theorems. A realistic picture of the discovery process looks like a small optimization problem. We surround a target statement with lemmas and theorems we already know, stretch them a little, introduce auxiliary parameters, and see whether they can be made to meet in the middle. From the blur perspective each of these “stretches” is a small blur parameter. We temporarily allow a theorem to be used in a slightly softened form—with extra error terms, extra parameters, or extra hypotheses—and assemble a provisional argument under these blurred rules. The resulting program is not yet a proof; it is a candidate that carries a cloud of blur parameters. The cleanup phase is then a systematic unsmoothing: remove parameters one by one, tighten inequalities, turn heuristic edges into crisp lemmas. The parameters that cannot be removed without breaking the argument are precisely the places where new information is needed. Sometimes that new information is a new lemma that we then prove; sometimes it is evidence that the current theory simply cannot see the problem cleanly and must be extended. This also explains why many successful proofs look, in retrospect, like the original theory with only a few new parameters or weighted versions added: those parameters were the blur knobs that could be turned off at the end. Additivity of blur The final theme is that blur from different sources tends to combine into a single budget. In the Collatz work, for example, many distinct contributions to ignorance—local variations in the accelerated map, truncation of tails, crude bounds on logarithms, combinatorial approximations— all collapse into one drift budget in the Lyapunov inequality. Once they are all expressed as small negative contributions, they simply add, and the combined drift is what matters. This additivity is not accidental. Unknown or ignored pieces of structure look alike once we step back far enough: they all live as small, signed error terms in inequalities that we can track. 7 Blur provides the language and the bookkeeping rule: name each source, add them into a single parameter, and make sure that parameter really can be pushed to zero or bounded in a way that the theorem can tolerate. Conclusion In all three cases—equidistribution, CLT, ergodic theorem—the sharp problem is subtle, but the blurred version is trivial: positivity, normalization, and approximate identity suffice. The same phenomenon reappears in ordinary limits and in the Riemann/Lebesgue comparison: blurring turns the problem into one about positive kernels and tail estimates, and the hard work is pushed into the unsmoothing step. Viewed from the broader epistemic angle, this is not an accident. Blur is the honest way to live with the thin band between what a theory already controls and what is still opaque. At the additive–multiplicative interface, at the finite–infinite seam, and in the process of discovery, we repeatedly accept small, structured losses of information in exchange for stability of the invariants we care about. The technical message of this note is that a single two-move template—blur, then unsmooth— organizes a large class of analytic arguments. The conceptual message is that this template mirrors how mathematics itself grows: we work at a controlled blur, peel off one stable piece of structure at a time, and leave the rest as a future budget. References [1] A. Perišić, Blur as a Universal Principle: Number Theory, Probability, Dynamics, Zenodo, 2025. [2] A. Perišić, Soft-Addition and Soft-Multiplication and the Channel–Switch Error, Zenodo, 2025. [3] A. Perišić, Blur at the Finite–Infinite Interface, Zenodo, 2025. [4] A. Perišić, Epistemological Blur, Zenodo, 2025. [5] A. Perišić, Randomness, Blur, and the □Operator, Zenodo, 2025. [6] A. Perišić, The Axiom of Blurred Choice, Zenodo, 2025. [7] A. Perišić, Multiplicative Choice, Zenodo, 2025. [8] A. Perišić, Randomness, Blur, and the Operator, Zenodo, 2025. [9] A. Perišić, A Lyapunov Certificate for the Accelerated Collatz Map, Zenodo, 2025. 8