arXiv subjects

Michael Jaber

Publications and source records attributed to Michael Jaber.

5 recordsLinked to original sources

arXiv2026

Covering $\mathbb{F}_2^n$ with Hamming Balls

Green asked the following question concerning structures in sumsets: Suppose that $\mathbb{F}_2^n$ is partitioned into sets $A_1, \dots, A_K$. Does $A_i+A_i$ contain a coset of codimension $O_K(1)$ for some $i$? An answer is not known even in the case of $K = 3$. We resolve this question in the affirmative in two special cases: (1) when $A_1, A_2$ are Hamming balls of radius $r < n/2 - 7$ relative to different bases, and (2) when $A_1$ is a Hamming ball of sufficiently small constant density.

math.CO↗

arXiv2025

Linear Hashing Is Optimal

We prove that hashing $n$ balls into $n$ bins via a random matrix over $\mathbf{F}_2$ yields expected maximum load $O(\log n / \log \log n)$. This matches the expected maximum load of a fully random function and resolves an open question posed by Alon, Dietzfelbinger, Miltersen, Petrank, and Tardos (STOC '97, JACM '99). More generally, we show that the maximum load exceeds $r\cdot\log n/\log\log n$ with probability at most $O(1/r^2)$.

cs.DS↗

arXiv2025

Quasipolynomial bounds for the corners theorem

Let $G$ be a finite abelian group and $A$ be a subset of $G \times G$ which is corner--free, meaning that there are no $x, y \in G$ and $d \in G \setminus \{0\}$ such that $(x, y)$, $(x+d, y)$, $(x, y+d) \in A$. We prove that \[|A| \le |G|^2 \cdot \exp(-(\log |G|)^{\Omega(1)}).\] As a consequence, we obtain polynomial (in the input length) lower bounds on the nondeterministic communication complexity of Exactly-N in the 3-player Number-on-Forehead model. We also obtain the first "reasonable'' lower bounds on the coloring version of the $3$-dimensional corners problem, as well as on the nondeterministic communication complexity of Exactly-N in the 4-player Number-on-Forehead model.

math.CO↗

arXiv2024

Corners in Quasirandom Groups via Sparse Mixing

We improve the best known upper bounds on the density of corner-free sets over quasirandom groups from inverse poly-logarithmic to quasi-polynomial. We make similarly substantial improvements to the best known lower bounds on the communication complexity of a large class of permutation functions in the 3-player Number-on-Forehead model. Underpinning both results is a general combinatorial theorem that extends the recent work of Kelley, Lovett, and Meka (STOC'24), itself a development of ideas from the breakthrough result of Kelley and Meka on three-term arithmetic progressions (FOCS'23).

math.CO↗

arXiv2024

Strong Bounds for Skew-Corner-Free Sets

Motivated by applications to matrix multiplication algorithms, Pratt asked (ITCS'24) how large a subset of $[n] \times [n]$ could be without containing a skew-corner: three points $(x,y), (x,y+h),(x+h,y')$ with $h \ne 0$. We prove any skew corner-free set has size at most $\exp(-\Omega(\log^{1/12} n))\cdot n^2$, nearly matching the best known lower bound of $\exp(-O(\sqrt{\log n}))\cdot n^2$ by Beker (arXiv'24). Our techniques generalize those of Kelley and Meka's recent breakthrough on three-term arithmetic progression (FOCS'23), answering a question of Beker (arXiv'24). We note that a similar bound was obtained concurrently and independently by Mili\'cevi\'c (arXiv'24).

math.CO↗