Parity problem

Abstract: For any m≥1, let H m denote the quantity . A celebrated recent result of Zhang showed the finiteness of H1, with the explicit bound H1≤70,000,000. This was then improved by us (the Polymath8 project) to H1≤4680, and then by Maynard to H1≤600, who also established for the first time a finiteness result for H m for m≥2, and specifically that H m ≪m3e4m. If one also assumes the Elliott-Halberstam conjecture, Maynard obtained the bound H1≤12, improving upon the previous bound H1≤16 of Goldston, Pintz, and Yildirim, as well as the bound H m ≪m3e2m. In this paper, we extend the methods of Maynard by generalizing the Selberg sieve further and by performing more extensive numerical calculations. As a consequence, we can obtain the bound H1≤246 unconditionally and H1≤6 under the assumption of the generalized Elliott-Halberstam conjecture. Indeed, under the latter conjecture, we show the stronger statement that for any admissible triple (h1,h2,h3), there are infinitely many n for which at least two of n+h1,n+h2,n+h3 are prime, and also obtain a related disjunction asserting that either the twin prime conjecture holds or the even Goldbach conjecture is asymptotically true if one allows an additive error of at most 2, or both. We also modify the ‘parity problem’ argument of Selberg to show that the H1≤6 bound is the best possible that one can obtain from purely sieve-theoretic considerations. For larger m, we use the distributional results obtained previously by our project to obtain the unconditional asymptotic bound or H m ≪m e2m under the assumption of the Elliott-Halberstam conjecture. We also obtain explicit upper bounds for H m when m=2,3,4,5.

Abstract: In [2], Blum et al. demonstrated the first sub-exponential algorithm for learning the parity function in the presence of noise. They solved the length-n parity problem in time 2O(n/logn) but it required the availability of 2O(n/logn) labeled examples. As an open problem, they asked whether there exists a 2o(n) algorithm for the length-n parity problem that uses only poly(n) labeled examples. In this work, we provide a positive answer to this question. We show that there is an algorithm that solves the length-n parity problem in time 2O(n/loglogn) using n1+e labeled examples. This result immediately gives us a sub-exponential algorithm for decoding n × n1+e random binary linear codes (i.e. codes where the messages are n bits and the codewords are n1+e bits) in the presence of random noise. We are also able to extend the same techniques to provide a sub-exponential algorithm for dense instances of the random subset sum problem.

Abstract: In (2), Blum et al. demonstrated the first sub-exponential algorithm for learning the parity function in the presence of noise. They solved the length-n parity problem in time 2 O(n/ log n) but it required the availability of 2 O(n/ log n) labeled examples. As an open problem, they asked whether there exists a 2 o(n) algorithm for the length-n parity problem that uses only poly(n) labeled examples. In this work, we provide a positive answer to this question. We show that there is an algorithm that solves the length-n parity problem in time 2 O(n/ log log n) using n 1+� labeled examples. This result immediately gives us a sub-exponential algorithm for decoding n × n 1+� random binary linear codes (i.e. codes where the messages are n bits and the codewords are n 1+� bits) in the presence of random noise. We are also able to extend the same techniques to provide a sub-exponential algorithm for dense instances of the random subset sum problem.