Sylvester-Gallai for Arrangements of Subspaces

In this work we study arrangements of $k$-dimensional subspaces $V_1,ldots,V_n subset mathbb{C}^ell$. Our main result shows that, if every pair $V_{a},V_b$ of subspaces is contained in a dependent triple (a triple $V_{a},V_b,V_c$ contained in a $2k$-dimensional space), then the entire arrangement must be contained in a subspace whose dimension depends only on $k$ (and not on $n$). The theorem holds under the assumption that $V_a cap V_b = {0}$ for every pair (otherwise it is false). This generalizes the Sylvester-Gallai theorem (or Kellys theorem for complex numbers), which proves the $k=1$ case. Our proof also handles arrangements in which we have many pairs (instead of all) appearing in dependent triples, generalizing the quantitative results of Barak et. al. [BDWY-pnas]. One of the main ingredients in the proof is a strengthening of a Theorem of Barthe [Bar98] (from the $k=1$ to $k>1$ case) proving the existence of a linear map that makes the angles between pairs of subspaces large on average. Such a mapping can be found, unless there is an obstruction in the form of a low dimensional subspace intersecting many of the spaces in the arrangement (in which case one can use a different argument to prove the main theorem).

Matching-Vector Families and LDCs Over Large Modulo

We prove new upper bounds on the size of families of vectors in $Z_m^n$ with restricted modular inner products, when $m$ is a large integer. More formally, if $vec{u}_1,ldots,vec{u}_t in Z_m^n$ and $vec{v}_1,ldots,vec{v}_t in Z_m^n$ satisfy $langlevec{u}_i,vec{v}_irangleequiv0pmod m$ and $langlevec{u}_i,vec{v}_jrangle otequiv0pmod m$ for all $i eq jin[t]$, we prove that $t leq O(m^{n/2+8.47})$. This improves a recent bound of $t leq m^{n/2 + O(log(m))}$ by cite{BDL13} and is the best possible up to the constant 8.47 when $m$ is sufficiently larger than $n$. The maximal size of such families, called `Matching-Vector families, shows up in recent constructions of locally decodable error correcting codes (LDCs) and determines the rate of the code. Using our result we are able to show that these codes, called Matching-Vector codes, must have encoding length at least $K^{19/18}$ for $K$-bit messages, regardless of their query complexity. This improves a known super linear bound of $ K2^{Omega({sqrt{log K}})}$ proved in cite{DGY11}.