Wirsching (2003) reduces uniform positive predecessor density for the $3n+1$ map to a chain of five conditions, organized into three conjectures. We prove two of them. Conjecture 1 concerns his path-counting generators: in the convolution carrying them to his Elka functions the partition weights grow subexponentially while the binomial ratio decays geometrically, so a window of width $O(\sqrt{\ell})$ dominates and its radius fits inside the hypothesis. Conjecture 3 concerns the asymptotics near 0 of an invariant density $\varphi$, a base-3 analogue of the Fabius density, against an explicit $\varphi_0$ due to Berg and Kr\"uppel. The exact log-Laplace transform of $\varphi$ splits into a smooth part, a log 3-periodic correction $H$, and a doubly exponentially small remainder. Berg and Kr\"uppel represented that correction as an infinite product in 1998; their analysis did not determine whether it is constant. We give $H$ as a Fourier series with coefficients in closed form in $\Gamma$ and $\zeta$; a classical zero-free theorem for $\zeta$ shows it is not constant, and we enclose its oscillation rigorously. Wirsching's comparison class fixes one phase of $H$, and there Conjecture 3 holds with limit $e^{H(0)}$, certified to lie in $(0.53412203666478,0.53412203666479)$. Off that class the phase sweeps a full period, so the unrestricted asymptotic $\varphi(t)\sim\kappa\varphi_0(t)$ fails. Condition $(\star4)$ follows, at every window radius, with $\mu=1/3$: what Wirsching's argument needs is weaker than Conjecture 3 itself, and the same saddlepoint chain settles it directly. With Conjecture 1 the chain reduces to the single condition $(\star3)$. Conjecture 2 is his route to it and remains open.
We prove Breiman's conjecture under the first-moment assumption. Let $Y_1,Y_2,\ldots$ be iid nonnegative random variables with $\mathbb P\{Y_1>0\}>0$, normalized by their sum. If the resulting randomly weighted sum converges to a nondegenerate law for one fixed integrable, nonconstant mark distribution, then the tail of $Y_1$ is regularly varying. More generally, any full-sequence limit for one such mark, including a constant limit, determines the asymptotic regime of the ranked weights: one big jump, a Poisson$\unicode{x2013}$Dirichlet partition, or dust. It consequently determines the limit for every integrable mark, with convergence in the $1$-Wasserstein metric, and the limits of independently marked empirical measures. The inverse step is based on a countable power-sum theorem: signed Fourier$\unicode{x2013}$Mellin identities extract a positive limiting expected power sum from one nondegenerate marked limit, without a moment of order greater than one. A ratio$\unicode{x2013}$Tauberian argument then recovers the tail index. The same method classifies ratios formed from the marked jumps of a nonzero, unkilled, driftless subordinator at zero and at infinity, assuming infinite activity at zero. A nondegenerate limit is equivalent to regular variation of the L\'evy tail with index in $(-1,0]$. A constant limit is equivalent to disappearance of the largest normalized jump, or, analytically, to slow variation of the integrated L\'evy tail. The latter condition need not imply regular variation of the L\'evy tail with index $-1$. A Cauchy-mark example shows that the conclusion can fail without integrability of the mark.
Fraenkel's conjecture asserts that a partition of the integers into at least three Beatty sequences with distinct moduli has the binary densities $1,2,4,\ldots,2^{m-1}$, normalized by $2^m-1$. We prove the conjecture through a dimension-free intermediate statement: every such partition contains a component of density at least 1/3. After reducing the partition to primitive common-period data, Fourier cancellation produces a finite inverse-sine system. We prove that no such system can exist when every density is below 1/3. The proof combines a divisor-concentration identity with uniform analytic estimates and three exact finite verifications, all carried out with integer or rational arithmetic. The component supplied by the density bound has mean spacing at most three. Deleting it preserves balance, and every surviving periodic balanced set is again a rational Beatty set. Induction determines the surviving binary scales, while a two-sequence disjointness criterion forces the deleted density to be the next binary scale. This yields the asserted density pattern.
Let $g_n$ be the largest number of Euclidean balls of diameter $1$ which may be needed to cover a set of diameter $1$ in $\mathbb{R}^n$. We study this problem for finite sets invariant under all coordinate permutations. We prove that the exponential growth rate in this symmetric problem can be characterized exactly as a finite-alphabet squared-error rate-distortion supremum $\alpha_0$. Specialized to the two-point case, i.e., for subsets of Boolean cubes, this gives the explicit lower bound \[g_n\ge (1.160235457\ldots-o(1))^n,\] improving the previous best bound $(2/\sqrt3-o(1))^n$. Using Fix's Gaussian characterization of the rate-distortion problem, we give a numerical three-point construction with exponent base greater than $1.160497831$. Finally, we show that $\alpha_0$ is not attained by any finitely supported distribution.
Andrii Arman, A. Bondarenko, A. Prymak et al.· 0 citations
Let $R_n$ be the infimum of the inradii of $\{z:|p(z)|<1\}$ over monic degree-$n$ polynomials whose zeros lie in the closed unit disk. We prove $nR_n\to\pi/2$, matching the asymptotic obstruction supplied by $z^n-1$. We first establish the exact universal radius $2^{1/n}-1$ for disks centered at zeros, which recovers the $(\log2)/n$ bound. Small inradius then forces radial concentration of the zeros and decay of their low reciprocal moments. These estimates give an entire limit with a modulus reflection identity; a second rescaling produces an exponential tangent and strict sublevel disks of every radius below $\pi/2$. The proof is accompanied by a Lean~4 formalization and a step-by-step source index.
Let $p$ be a complex polynomial of degree $n\ge2$ whose zeros lie in the closed unit disk. Sendov's conjecture asserts that every zero of $p$ lies within distance one of a zero of $p'$. Mazur's recent proof, and Tao's streamlined exposition of it, reduce a hypothetical counterexample to a scalar lower bound \[ 1\le F(\eta,\alpha,n) \] together with two upper bounds for the endpoint parameter $\eta$ and the uniform restriction $0<\alpha\le 17$. We complete this scalar reduction by a two-stage optimization argument. First, $F$ is nondecreasing in $\eta$, so $\eta$ may be replaced by a piecewise polar envelope $\eta^{\ast}(\alpha)$. Writing $s=(n-1)/2$ converts the degree to a half-integer variable and gives \[ F(\eta^{\ast}(\alpha),\alpha,n) = E(\alpha,s) + \kappa(\alpha,s) \int_0^1 t^3 \widehat{\beta}(t;\alpha,s)^{\,s-\frac{3}{2}} \,dt. \] On each half-unit $\alpha$-slab, $E$ and $\kappa$ decrease with $\alpha$, whereas $\widehat\beta$ increases. Convexity in $t$ reduces all but the first mesh interval to eight explicit nodal terms. The first interval satisfies a uniform bound $3/200$. Each nodal upper term is a strictly log-concave function of the half-integer $s$, so its global discrete maximum is certified by two adjacent ratio evaluations. A fixed finite certificate over the $34$ half-unit slabs gives \[ F(\eta^{\ast}(\alpha),\alpha,n)<\frac{97}{100}<1, \] contradicting the scalar lower bound. The main contribution is a substantial reduction of the computer assistance required after the Mazur--Tao scalar reduction: all continuous optimization and the unbounded degree parameter are handled analytically, leaving only a small fixed collection of explicit one-variable inequalities.
Let $\varphi$ be a minimal counterexample to Vaught's conjecture in the sense of Montalban; such a $\varphi$ exists if Vaught's conjecture fails, by Steel and Harnik-Makkai. We bring into contact, at the level of statements, two bodies of work on the models of such a $\varphi$: the analysis of Gonzalez-Rossegger-Turetsky, in which at every countable level $\beta$ exactly one back-and-forth class $C_\beta$ is uncountable and, at fixed points of an associated function, has a distinguished member of least Scott rank (its label); and the supply of models with prescribed $\omega_1^A$ from higher recursion theory, namely Montalban's Gandy-basis lemma and Sacks'$\Sigma_1$-hull club. All results are theorems of ZFC under standing hypotheses (H0)-(H3). We prove: (i) a per-witness trichotomy -- for every limit $\lambda$ in the fixed-point club above $qr(\varphi)$, every model $A$ of $\varphi$ with $\omega_1^A=\lambda$ and Scott rank at least $\lambda$ either is the label $K_\lambda$, or lies outside $C_{\lambda+1}$ and so forces two non-isomorphic models of Scott rank $\lambda+1$, or is the label $K_{\lambda+1}$ and attains the Nadel bound; (ii) coordinate identities: the first and third branches are equivalent, level by level, to computations of $\omega_1$ of the labels; (iii) a seeding theorem: on a club, Sacks'construction supplies at every level a model of top rank with prescribed $\omega_1$, and his atomic chain consists of the labels; (iv) an unconditional stationary dichotomy: on that club, either stationarily many successor levels carry two non-isomorphic models of Scott rank $\lambda+1$, or stationarily many labels attain the Nadel bound. On the fiber of models with $\omega_1=\lambda$ we further prove uniqueness of the node-saturated model and give a complete isomorphism invariant with countable spectrum. We prove nothing bearing on Vaught's conjecture itself.
M. Mahmoud· 0 citations
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