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Statement

Setting (pp. 2, 4, 6). The $r_i(t)=(-1)^{[2^it]}$, $t\in[0,1]$, are the Rademacher functions, and $(r_i\otimes r_j)(u,v)=r_i(u)r_j(v)$ on $[0,1]^2$. The expectation $\mathsf E_\theta$ is over all arrangements of signs $\theta_{i,j}=\pm1$. In the paper, $F_1\asymp F_2$ means $cF_1\le F_2\le CF_1$ for constants $c,C>0$ that do not depend on all or part of the arguments of $F_1$ and $F_2$ (p. 4).

Theorem 2 (p. 14). There are universal constants such that, for all $n,m\in\mathbb N$ and all real $a_{i,j}$, $1\le i\le n$, $1\le j\le m$,

$$\mathsf E_\theta\Bigl\|\sum_{i=1}^n\sum_{j=1}^m\theta_{i,j}a_{i,j}\,r_i\otimes r_j\Bigr\|_{L_\infty([0,1]^2)} \asymp\min_{\theta_{i,j}=\pm1}\Bigl\|\sum_{i=1}^n\sum_{j=1}^m\theta_{i,j}a_{i,j}\,r_i\otimes r_j\Bigr\|_{L_\infty([0,1]^2)}$$ $$\asymp\max\Bigl\{\sum_{i=1}^n\Bigl(\sum_{j=1}^ma_{i,j}^2\Bigr)^{1/2},\ \sum_{j=1}^m\Bigl(\sum_{i=1}^na_{i,j}^2\Bigr)^{1/2}\Bigr\}.$$

In the terms of Definition 1 (p. 6), the system $\{r_i\otimes r_j\}$ is a system of random unconditional convergence (an RUC system) in $L_\infty([0,1]^2)$; the paper records this as Corollary 2 (p. 16). The $L_\infty$ norm of the sum equals the norm of the matrix $(a_{i,j})$ as an operator from $\ell_\infty^m$ to $\ell_1^n$ (equation (10), p. 9), and lies between the cut-norm $\|(a_{i,j})\|_{cut}$ of equation (11) and four times it (equation (13), p. 10, from Alon and Naor). So the same three-way equivalence holds with the cut-norm of $(\theta_{i,j}a_{i,j})$ in place of the $L_\infty$ norm; that is Corollary 3 (p. 16), the input to Theorem 5.

Source. Sergey V. Astashkin and Konstantin V. Lykov, Random unconditional convergence of Rademacher chaos in $L_\infty$ and sharp estimates for discrepancy of weighted graphs and hypergraphs, arXiv:2412.20107v1 [math.PR], 28 December 2024; Section 3 (pp. 13--16), Theorem 2 on p. 14, its proof on pp. 14--16. The edition read is identified on the source card.

Read depth. Claims checked: the statement, its hypotheses and the notation it uses were read clause by clause on the page images. The proof was read for its structure, summarized below, and not checked step by step. Nothing here is independently reviewed.

Proof pointer

Pp. 14--16. The lower bounds come from Lemma 1 (p. 14): for any coefficients the $L_\infty$ norm is at least $1/\sqrt2$ times the larger mixed sum, by Szarek's inequality with its sharp constant in $L_1$. Since the minimum over signs is at most the average, this bounds both from below. The upper bound for the average is Lemma 2 (p. 15): by equation (10) the average is the expected operator norm, from $\ell_\infty^m$ to $\ell_1^n$, of the random sign matrix $(r_{i,j}a_{i,j})$, and this is at most a universal constant times the larger mixed sum. Its proof bounds the Gaussian version through Proposition 1.8(i) of Adamczak, Prochno, Strzelecka and Strzelecki (Math. Ann. 388 (2024)) and passes from Gaussian to Rademacher sums with the factor $\sqrt{\pi/2}$.

Dependencies

Within the paper: Lemmas 1 and 2 (pp. 14--15) and equation (10) (p. 9). Outside it: Szarek's inequality (Studia Math. 58 (1976)) and the cited proposition of Adamczak, Prochno, Strzelecka and Strzelecki, used as stated.

Bears on

No Erdős problem directly. Through Corollary 3 (p. 16) it gives Theorem 5 on weighted complete bipartite graphs, and its upper bound is used in the proof of Theorem 3.