# Vector Balancing in Lebesgue Spaces

@article{Reis2022VectorBI, title={Vector Balancing in Lebesgue Spaces}, author={Victor Reis and Thomas Rothvoss}, journal={ArXiv}, year={2022}, volume={abs/2007.05634} }

A tantalizing conjecture in discrete mathematics is the one of Komlos, suggesting that for any vectors $\mathbf{a}_1,\ldots,\mathbf{a}_n \in B_2^m$ there exist signs $x_1, \dots, x_n \in \{ -1,1\}$ so that $\|\sum_{i=1}^n x_i\mathbf{a}_i\|_\infty \le O(1)$. It is a natural extension to ask what $\ell_q$-norm bound to expect for $\mathbf{a}_1,\ldots,\mathbf{a}_n \in B_p^m$. We prove that, for $2 \le p \le q \le \infty$, such vectors admit fractional colorings $x_1, \dots, x_n \in [-1,1]$ with a…

## 4 Citations

### A new framework for matrix discrepancy: partial coloring bounds via mirror descent

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The matrix Spencer conjecture is reduced to the existence of a O(log(m/n) quantum relative entropy net on the spectraplex using a Gaussian measure lower bound of 2−O(n) for a scaling of the discrepancy body.

### Improved Algorithms for Combinatorial Discrepancy

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Discrepancy theory is a subfield of combinatorics which has branched in Computer Science due to several connections it has to geometric problems, randomized algorithms and complexity theory [13, 8].…

### Hyperbolic Concentration, Anti-Concentration, and Discrepancy

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This paper presents the first work that shows either concentration or anti-concentration results for hyperbolic polynomials and implies that a random bi-coloring of any set system with n sets and n elements will have discrepancy O ( √ n log n ) with high probability.

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A simple proof of the matrix Spencer conjecture up to poly-logarithmic rank is given, which implies a log n − Ω(log log n ) qubit lower bound for quantum random access codes encoding n classical bits with advantage ≫ 1 / √ n .

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