For a compact set $K\subset\mathbb{C}$, let $\vartheta(K)$ be the infimum of the planar areas of the unit lemniscates of all monic polynomials with zeros in $K$, allowing arbitrary degree and repeated zeros. We prove that $\vartheta(K)=0$ whenever $\operatorname{cap}(K)=1$, with no regularity assumption on $K$. The proof uses a centered harmonic polynomial that is positive on all but a set of arbitrarily small area in the polynomial hull of $K$. A Fourier average of exterior harmonic measures realizes this polynomial as the logarithmic potential of a signed measure having bounded density with respect to the equilibrium measure. A positive perturbation and an $L^1$ approximation by empirical measures then produce the required polynomials. This extends the smooth-boundary result of Krishnapur, Lundberg, and Ramachandran to arbitrary compact sets of capacity one. Together with the capacity-greater-than-one theorem of Ghosh and Ramachandran and an elementary argument for unbounded sets, it follows that $\vartheta(F)=0$ for every closed infinite set $F\subset\mathbb{C}$ of transfinite diameter at least one, answering the vanishing question in Erd\H{o}s Problem 1040.
Let $A\subseteq\mathbb{R}$ support a probability measure whose Fourier--Stieltjes transform tends to zero at infinity. We prove that, for every $\varepsilon\in(0,1)$, there is a closed, $1$-periodic, nowhere dense set $E\subseteq\mathbb{R}$ such that \[ m(E\cap I)\ge1-\varepsilon \] for every interval $I$ of length $1$...
A. Iosevich, N. Kulkarni, N. Cuéllar et al.· 0 citations
Suppose a set of integers $A\subseteq\{1,\dots,N\}$ has no solutions to $a-a'=n\lfloor\sqrt[3]2n\rfloor,$ for distinct $a,a'\in A,$ and $n\in \mathbb{N}.$ We show that $|A|\ll N^{1-c}$ for some absolute constant $c>0.$ To do this, we prove quantitative bounds on the van der Corput property for certain sets of bracket q...
Let ${R^\ast}(n)$ be the least radius of a central circle to which nonoverlapping circles of radii $1,\ldots,n$ are externally tangent. We prove that ${R^\ast}(n)={C_\ast} n^2+o(n^2)$ and characterize ${C_\ast}$ by finite linear programs with an explicit error tending to zero. The reduction preserves arbitrary orders a...
Let $n,d\geq1$, and suppose that $E\subset \mathbb{Q}_p^n$ contains a degree-$d$ polynomial image of $\mathbb{Z}_p$ in every leading direction of a nonempty set $V\subset\mathbb{P}^{n-1}(\mathbb{Q}_p)$. We prove the sharp bound $\dim_H E\geq\dim_H V+1$. For joint block-valued polynomial evaluations, the total incidence...
Let $(a_n)$ be a positive null sequence with $\liminf_{n\to\infty}a_{n+1}/a_n>0$. For every $\eta>0$, we construct a compact set $E\subset[0,1]$ with Lebesgue measure greater than $1-\eta$ containing no nontrivial affine copy of $\{a_n:n\ge1\}$. No monotonicity is assumed. The proof adapts a finite-tree construction fo...
Let $B^d$ denote the $d$-dimensional Euclidean ball of unit radius. What is the largest constant $f(B^d) \in [0,1]$ with the property that every finite collection $\mathcal{C}$ of unit balls in $\mathbb{R}^d$ admits a disjoint sub-collection $\mathcal{S}$ occupying at least a fraction $f(B^d)$ of the volume of $\mathca...
Machine-learning networks can learn without detailed instructions from a designer. Those that do may offer insight into how the brain reorganizes itself during learning.
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