In 1997, Belinsky conjectured that, for convex subsequences, the logarithmic growth condition of Carleson, Trigub, and Zagorodni\u{\i} is necessary and sufficient for the arithmetic means of subsequential Fourier partial sums to converge at every Lebesgue point of every integrable function. We disprove the sufficiency part of this conjecture. More precisely, we construct a strictly convex increasing sequence $(a_m)$ satisfying $a_m\leq 7m^8$ and a function $f\in L^1(\mathbb T)$ for which $0$ is a Lebesgue point, $f(0)=0$, and the means $m^{-1}\sum_{k=1}^m S_{a_k}f(0)$ are unbounded.
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 $m\geq 2$, let $c_p(m)$ be the all-dimensional best constant in $$ \left\|\sum_{k=1}^m A_k\right\|_p \leq c_p(m)\left\|\sum_{k=1}^m |A_k|\right\|_p. $$ Tang and Zhang conjectured an explicit formula for every finite $p>1$. We disprove the conjecture with two explicit real $2\times 2$ rank-one matrices at $p=3/2$. T...
Very recently, Lech Mazur proved the celebrated Sendov conjecture, and Terence Tao subsequently distilled the main ideas of the proof in a blog post. In this paper, we establish a quantitative strengthening of Sendov's conjecture, namely the quadratic Tang--Zhang inequality. Let $p$ be a polynomial of degree $n\ge2$ wh...
We prove the Zygmund conjecture in all parameters: the maximal operator associated with products of cubes with side lengths $(s_1,\ldots,s_{m-1},\phi(s_1,\ldots,s_{m-1}))$, where $\phi$ is any positive coordinatewise nondecreasing function, satisfies the weak $L(\log L)^{m-2}$ estimate for every $m\geq3$. The estimate...
In 1936, Zalcwasser proved the almost everywhere convergence of the arithmetic means of the square subsequence of trigonometric Fourier partial sums and asked whether this result extends to higher powers and to Ces\`aro means of fractional order. We give affirmative answers to both questions in a stronger pointwise for...
The Schiffer conjecture states that if a smooth domain $\Omega \subset \mathbb{R}^n$ admits a Neumann eigenfunction of the Laplacian which is constant at the boundary, then the domain is a ball. It is intimately related to Pompeiu's problem, stating that if a nonzero function integrates zero over any rigid motion of $\...
G. Cao-Labora, Jaume de Dios Pont· 3 citations
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