Abstract
We examine the behavior of the number of k-term arithmetic progressions in a random subset of ℤ/nℤ. We prove that if a set is chosen by including each element of ℤ/nℤ independently with constant probability p, then the resulting distribution of k-term arithmetic progressions in that set, while obeying a central limit theorem, does not obey a local central limit theorem. The methods involve decomposing the random variable into homogeneous degree d polynomials with respect to the Walsh/Fourier basis. Proving a suitable multivariate central limit theorem for each component of the expansion gives the desired result.
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References
Y. Barhoumi-Andréani, C. Koch and H. Liu, Bivariate fluctuations for the number of arithmetic progressions in random sets, Electronic Journal of Probability 24 (2019), Article no. 145.
R. Berkowitz, A local limit theorem for cliques in G(n, p), https://arxiv.org/abs/1811.03527.
R. Berkowitz, A quantitative local limit theorem for triangles in random graphs, https://arxiv.org/abs/1610.01281.
B. B. Bhattacharya, S. Ganguly, X. Shao and Y. Zhao, Upper tail large deviations for Arithmetic progressions in a random set, International Mathematics Research Notices 1 (2020), 167–213.
B. Cai, A. Chen, B. Heller and E. Tsegaye, Limit theorems for descents in permutations and Arithmetic progressions in ℤ/pℤ, https://arxiv.org/abs/1810.02425.
J. Fox, M. Kwan and L. Sauermann, Anticoncentration for subgraph counts in random graphs, Annals of Probability, to appear, https://arxiv.org/abs/1905.12749.
J. Gilmer and S. Kopparty, A local central limit theorem for triangles in a random graph, Random Structures & Algorithms 48 (2016), 732–750.
M. Harel, F. Mousset and W. Samotij, Upper tails via high moments and entropic stability, https://arxiv.org/abs/1904.08212.
S. Janson and L. Warnke, The lower tail: Poisson approximation revisited, Random Structures & Algorithms 48 (2016), 219–246.
E. Meckes, On Stein’s method for multivariate normal approximation, in High Dimensional Probability. V: The Luminy Volume, Institute of Mathematical Statistics Collections, Vol. 5, Institute of Mathematical Statistics, Beachwood, OH, 2009, pp. 153–178.
R. O’Donnell, Analysis of Boolean Functions, Cambridge University Press, New York, 2014.
L. Warnke, Upper tails for arithmetic progressions in random subsets, Israel Journal of Mathematics 221 (2017), 317–365.
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Berkowitz, R., Sah, A. & Sawhney, M. Number of arithmetic progressions in dense random subsets of ℤ/nℤ. Isr. J. Math. 244, 589–620 (2021). https://doi.org/10.1007/s11856-021-2180-7
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DOI: https://doi.org/10.1007/s11856-021-2180-7