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Statement

Setting as for Theorem 2: the $r_i$ are the Rademacher functions on $[0,1]$, $\mathsf E_\theta$ is the expectation over all arrangements of signs $\theta_{i,j}=\pm1$, $1\le i<j\le n$, and $\asymp$ is two-sided comparability up to constants (p. 4). The functions $r_ir_j$, $i<j$, form the second-order Rademacher chaos (p. 7).

Theorem 3 (p. 17). There are universal constants such that, for all $n\in\mathbb N$ and all real $a_{i,j}$, $1\le i<j\le n$,

$$\mathsf E_\theta\Bigl\|\sum_{i=1}^n\sum_{j=i+1}^n\theta_{i,j}a_{i,j}r_ir_j\Bigr\|_{L_\infty} \asymp\min_{\theta_{i,j}=\pm1}\Bigl\|\sum_{i=1}^n\sum_{j=i+1}^n\theta_{i,j}a_{i,j}r_ir_j\Bigr\|_{L_\infty}$$ $$\asymp\max\Bigl\{\sum_{i=1}^{n-1}\Bigl(\sum_{j=i+1}^na_{i,j}^2\Bigr)^{1/2},\ \sum_{j=2}^n\Bigl(\sum_{i=1}^{j-1}a_{i,j}^2\Bigr)^{1/2}\Bigr\}.$$

This is the equivalence (3) announced in the introduction (p. 2). The proof gives the explicit lower bound

$$\Bigl\|\sum_{i=1}^n\sum_{j=i+1}^na_{i,j}r_ir_j\Bigr\|_{L_\infty}\ge\frac1{16\sqrt2}\max\Bigl\{\sum_{i=1}^{n-1}\Bigl(\sum_{j=i+1}^na_{i,j}^2\Bigr)^{1/2},\ \sum_{j=2}^n\Bigl(\sum_{i=1}^{j-1}a_{i,j}^2\Bigr)^{1/2}\Bigr\}$$

for every choice of coefficients (equation (27), p. 18); the upper constant is the unspecified universal constant of Theorem 2.

With the modified cut-norm $\|A\|_{cut}^*=\max\{|\sum_{i,j\in I,\,i<j}a_{i,j}|:I\subset[n]\}$ of equation (18) (p. 11), which equation (19) (p. 11) shows to be equivalent to the $L_\infty$ norm of the chaos sum with constants independent of $n$ and $A$, the theorem yields Corollary 4 (p. 18): for each strictly upper triangular matrix $A=(a_{i,j})_{1\le i<j\le n}$, with universal constants, $\mathsf E_\theta\|(\theta_{i,j}a_{i,j})\|_{cut}^*$ and $\min_{\theta_{i,j}=\pm1}\|(\theta_{i,j}a_{i,j})\|_{cut}^*$ are both equivalent to the same maximum. Corollary 4 is the input to Theorem 6.

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 4 (pp. 16--18), Theorem 3 on p. 17, its proof on pp. 17--18, Corollary 4 on p. 18. The edition read is identified on the source card.

Read depth. Claims checked: the statement, Corollary 4 and the definitions of equations (14)--(19) were read clause by clause on the page images. The proof and the derivation of (19) were read for their structure, summarized below, and not checked step by step. Nothing here is independently reviewed.

Proof pointer

Pp. 17--18. For the lower bound, symmetrize: put $b_{i,j}=a_{i,j}/2$ for $i<j$, $b_{j,i}=b_{i,j}$ and $b_{i,i}=0$, so that $\sum_{i,j}b_{i,j}r_ir_j$ is the chaos sum. The decoupling inequality of Corollary 1 (p. 8) at $d=2$ bounds the decoupled sum $\sum_{i,j}b_{i,j}r_i\otimes r_j$ by four times the chaos sum in $L_\infty$, and Lemma 1 (p. 14) bounds the decoupled sum from below by the mixed sums of $(b_{i,j})$; comparing these with the two triangular sums gives (27). For the upper bound, the chaos sum is pointwise a restriction of the decoupled sum to the diagonal $u=v$, so its $L_\infty$ norm is at most the decoupled norm, and the upper bound of Theorem 2 applies to the array with signs $\theta_{i,j}a_{i,j}$ for $i<j$ and zero elsewhere.

Dependencies

Within the paper: Corollary 1 (p. 8, from the decoupling Theorem 1, p. 7, which the paper takes from de la Peña and Giné), Lemma 1 (p. 14) and Theorem 2 (p. 14). Corollary 4 further uses equation (19) (p. 11).

Bears on

No Erdős problem directly. Through Corollary 4 (p. 18) it gives Theorem 6 and Theorem 7, whose unit-weight case is the unordered-edge quantity of Problem 1028.