DeLeo, Henderschedt, and Wells introduced a finite-horizon version of the classical de Bruijn--Erdos interval discrepancy problem. Starting from the unit interval, one repeatedly splits an existing interval into two until $n$ intervals are present, and one minimizes the largest ratio between the longest and shortest intervals over all intermediate partitions. They constructed the lex-merge strategy, whose discrepancy is $2^{1-1/\lceil n/2\rceil}$, and conjectured that this value is optimal for every $n$. We prove the conjecture. More generally, we establish a sharp lower bound for arbitrary binary refinement processes of positive masses: any process that starts with one positive mass, repeatedly replaces one mass by two positive masses with the same total, and terminates with $n$ masses must at some stage have largest-to-smallest ratio at least $2^{1-1/\lceil n/2\rceil}$. The proof tracks the minimum mass under refinement and uses the forced survival of a piece near the midpoint of the process. We also record the corresponding universal lower bound for $r$-ary refinements.
We construct an infinite family of solutions to the Mondrian art problem with absolute defect bounded above by $280$, of side lengths $n_0 = 315$, $n_1 = 106785$, and $n_{j+2} = 340n_{j+1} - n_j$, where $n_j$ is congruent to $315 \pmod{630}$. This absolute bound shows that the minimum defect does not necessarily increa...
An influential 1989 result of Naddef shows that the diameters of $0/1$-polytopes are at most their dimension. This was extended shortly after by Kleinschmidt and Onn to any lattice polytope in $[0,k]^{d}$, where they showed a bound of at most $dk$. Naddef's argument easily extends to the monotone setting motivated by t...
For even $n$, let $P_n$ be the probability that independent uniform strict preference lists on $n$ participants admit a stable perfect matching. We prove \[ P_n\sim\frac{e\,2^{1/4}\Gamma(3/4)}{\sqrt\pi}\,n^{-1/4}. \] This establishes Mertens's conjectured exponent of decay, with a leading constant different from his or...
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 o...
For every positive integer $n$, we prove that there is a constant $C_n$, depending only on $n$, such that $Vol(-K_X) \leq C_n \epsilon^{-(2^n-n-1)}$ whenever $X$ admits an $\epsilon$-lc log Fano boundary. The exponent is optimal in every dimension at least two, even for toric Fano varieties of Picard number one. In par...
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 w...
Renato Augusto Tavares· 0 citations
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