We give an elementary proof of the Koml\'os conjecture by simplifying the recent proof of Guo, Fang, and Lu. We show that any vectors $v_1,\ldots,v_n\in\mathbb{R}^d$ with $\|v_i\|_2\le1$ admit signs $\varepsilon_i\in\{-1,1\}$ such that $\|\sum_{i=1}^n\varepsilon_i v_i\|_\infty\le36$. The proof uses only elementary combinatorial and probabilistic arguments and basic calculus.
Guo, Fang, and Lu recently proved the Koml\'os conjecture: for vectors $v_1,\ldots,v_n\in\mathbb R^d$ of Euclidean norm at most one, there are signs $\varepsilon_j\in\{-1,1\}$ with $\|\sum_j\varepsilon_jv_j\|_\infty\le3\sqrt{2\pi}$. We give a short proof of the bound $3\pi$ that keeps the geometric lifting framework of...
Let $\lambda_1(\Omega)$ and $T(\Omega)$ denote the first Dirichlet eigenvalue and torsional rigidity of a bounded convex domain $\Omega\subset\mathbb{R}^2$, and let $M=\max_\Omega u$, where $u$ is the torsion function. We prove the sharp inequalities $\pi^2/24<\lambda_1(\Omega)T(\Omega)/|\Omega|<\pi^2/12$. This proves...
Let $\boldsymbol{n}=(n_1,n_2,n_3):(\mathbb{C}^2,0)\to(\mathbb{C}^3,0)$ be a holomorphic map germ admitting an injective representative. We prove that if $(d\boldsymbol{n})_0=0$, then $\mathrm{ord}_0(\boldsymbol{n})\in\{2,3,4\}$. This reduces L\^e's conjecture to excluding potential counterexamples $\boldsymbol{n}$ of o...
We show that there exists $c>0$ such that any subset of $\{1,\ldots,N\}$ having size $\gg N / \exp( (\log\log\log N)^c )$ contains a nontrivial pattern of the form $x,x+y,x+2y,x+y^3$. It is the first configuration of complexity strictly greater than $0$, other than refinements of arithmetic progressions, for which quan...
Bartłomiej Bychawski, Bartosz Głowacki, Ivan Spyrydonov· 0 citations
By distilling and combining earlier arguments, we give a self-contained and entirely elementary proof of the strong and real linear polarization inequalities established by Mart\'inez and Ortega-Moreno. Specifically, for any unit vectors $u_1,\ldots,u_n\in\mathbb{R}^d$, there exists a unit vector $v\in\mathbb{R}^d$ suc...
We construct, for every $d\ge 5$, a nonnegative radial potential $V\in L^\infty_{\mathrm{loc}}(\mathbb{R}^d\setminus\{0\})\cap L^1(\mathbb{R}^d)$ for which $V(-\Delta\dotplus V)^{-1}$ is not bounded on $L^2(\mathbb{R}^d)$.
M. A. Perelmuter· 0 citations
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