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تسجيل مستخدم جديدSylvester-Gallai type theorems for approximate collinearity
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We study questions in incidence geometry where the precise position of points is `blurry (e.g. due to noise, inaccuracy or error). Thus lines are replaced by narrow tubes, and more generally affine subspaces are replaced by their small neighborhood. We show that the presence of a sufficiently large number of approximately collinear triples in a set of points in d dimensional complex space implies that the points are close to a low dimensional affine subspace. This can be viewed as a stable variant of the Sylvester-Gallai theorem and its extensions. Building on the recently found connection between Sylvester-Gallai type theorems and complex Locally Correctable Codes (LCCs), we define the new notion of stable LCCs, in which the (local) correction procedure can also handle small perturbations in the euclidean metric. We prove that such stable codes with constant query complexity do not exist. No impossibility results were known in any such local setting for more than 2 queries.
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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 $2k$-
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A (q,k,t)-design matrix is an m x n matrix whose pattern of zeros/non-zeros satisfies the following design-like condition: each row has at most q non-zeros, each column has at least k non-zeros and the supports of every two columns intersect in at mo
st t rows. We prove that the rank of any (q,k,t)-design matrix over a field of characteristic zero (or sufficiently large finite characteristic) is at least n - (qtn/2k)^2 . Using this result we derive the following applications: (1) Impossibility results for 2-query LCCs over the complex numbers: A 2-query locally correctable code (LCC) is an error correcting code in which every codeword coordinate can be recovered, probabilistically, by reading at most two other code positions. Such codes have numerous applications and constructions (with exponential encoding length) are known over finite fields of small characteristic. We show that infinite families of such linear 2-query LCCs do not exist over the complex numbers. (2) Generalization of results in combinatorial geometry: We prove a quantitative analog of the Sylvester-Gallai theorem: Let $v_1,...,v_m$ be a set of points in $C^d$ such that for every $i in [m]$ there exists at least $delta m$ values of $j in [m]$ such that the line through $v_i,v_j$ contains a third point in the set. We show that the dimension of ${v_1,...,v_m }$ is at most $O(1/delta^2)$. Our results generalize to the high dimensional case (replacing lines with planes, etc.) and to the case where the points are colored (as in the Motzkin-Rabin Theorem).
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