Anti-concentration in most directions
A Rao, A Yehudayoff - arXiv preprint arXiv:1811.06510, 2018 - arxiv.org
arXiv preprint arXiv:1811.06510, 2018•arxiv.org
We prove anti-concentration bounds for the inner product of two independent random
vectors. For example, we show that if $ A, B $ are subsets of the cube $\{\pm 1\}^ n $ with $|
A|\cdot| B|\geq 2^{1.01 n} $, and $ X\in A $ and $ Y\in B $ are sampled independently and
uniformly, then the inner product $\langle X, Y\rangle $ takes on any fixed value with
probability at most $ O (\tfrac {1}{\sqrt {n}}) $. Extending Hal\'asz work, we prove stronger
bounds when the choices for $ x $ are unstructured. We also describe applications to …
vectors. For example, we show that if $ A, B $ are subsets of the cube $\{\pm 1\}^ n $ with $|
A|\cdot| B|\geq 2^{1.01 n} $, and $ X\in A $ and $ Y\in B $ are sampled independently and
uniformly, then the inner product $\langle X, Y\rangle $ takes on any fixed value with
probability at most $ O (\tfrac {1}{\sqrt {n}}) $. Extending Hal\'asz work, we prove stronger
bounds when the choices for $ x $ are unstructured. We also describe applications to …
We prove anti-concentration bounds for the inner product of two independent random vectors. For example, we show that if are subsets of the cube with , and and are sampled independently and uniformly, then the inner product takes on any fixed value with probability at most . Extending Hal\'asz work, we prove stronger bounds when the choices for are unstructured. We also describe applications to communication complexity, randomness extraction and additive combinatorics.
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