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...via proving an exponential lower bound on the discrepancy of set systems of subcubes of the boolean cube {0, 1}d. 1. Introduction Circa 1932 Paul Erdős made the following conjecture: Conjecture 1.1 (=-=[2]-=-). For any function f : N → {−1,+1} and for any constant C, there exist positive integers n and a such that | ⌊n/a⌋∑ i=1 f(ia)| > C. In modern terms, the maximum value of ∑k i=1 f(ia) over all a and a...

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...homogeneous arithmetic progressions. We prove a lower bound on the set system of boolean subcubes using the determinant lower bound on hereditary discrepancy due to Lovász, Spencer, and Vesztergombi =-=[6]-=-. Such systems of boolean subcubes were previously considered in computer science in the context of private data analysis [5, 8]. Our construction produces sets Wn of square free integers with a large...

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... bound on hereditary discrepancy due to Lovász, Spencer, and Vesztergombi [6]. Such systems of boolean subcubes were previously considered in computer science in the context of private data analysis =-=[5, 8]-=-. Our construction produces sets Wn of square free integers with a large number of prime divisors, suggesting that such integers are a chief obstacle in achieving bounded discrepancy for homogeneous a...

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... boolean cube. Such set systems have applications in the theory of differential privacy. The proof of Lemma 3.2 together with the connection between discrepancy and differential privacy formalized in =-=[7]-=- can be used to give simpler proofs of the noise lower bounds of the type considered in [5]. It is an interesting question whether discrepancy bounds on set systems of boolean subcubes can find other ...

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... discrepancy problem recently received attention as the subject of the fifth polymath project [1]. Our note is motivated by recent results of Alon and Kalai, announced and sketched in the weblog post =-=[4]-=-. Using the Beck-Fiala theorem, they showed that even for homogeneous arithmetic progressions restricted to an arbitrary subset of the integers up to n, the discrepancy is no more than n1/Ω(log logn)....

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... bound on hereditary discrepancy due to Lovász, Spencer, and Vesztergombi [6]. Such systems of boolean subcubes were previously considered in computer science in the context of private data analysis =-=[5, 8]-=-. Our construction produces sets Wn of square free integers with a large number of prime divisors, suggesting that such integers are a chief obstacle in achieving bounded discrepancy for homogeneous a...

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...e f(i) = −1 if and only if the last nonzero digit of i in ternary representation is 2 has discrepancy O(log n). For references and other partial results related to the Erdős discrepancy problem, see =-=[3, 1]-=-. The Erdős discrepancy problem recently received attention as the subject of the fifth polymath project [1]. Our note is motivated by recent results of Alon and Kalai, announced and sketched in the ...