Cusps of primes in dense subsequences -- Bypassing the W-trick

2024/12/16 by Ramaré, Olivier · 1 citation
#11L07 #11L20 #11N36 #FOS: Mathematics #Number Theory (math.NT) #Primary: 11N05 #Secondary: 11P32

paper · doi:10.48550/arxiv.2412.11527

Abstract

Let the A-cusps of a dense subset P^*∈[√(N),N] of primes be points α∈ℝ/ℤ that are such that |∑_\substackp\inP^* e(αp)|≥ |P^*|/A. We establish that any (1/N)-well spaced subset of A-cusps contains at most 20A²Klog(2A) points, where K=N/(|P^*|log N). We further show that any B-cusps~ξ is accompanied, when B≤ √(A), by a large proportion of A-cusps of the shape ξ+(a/q). We conclude this study by showing that, given A≥2, the characteristic function 1_P^* may be decomposed in the form 1_P^*=(V(z₀)log N)^-1f^\flat +f^\sharp where the trigonometric polynomial of f^\sharp takes only values ≤ |P^*|/A, and~f^\flat is a bounded non-negative function supported on the integers prime to M; the parameters z₀ and M are given in terms of~A, while V(z₀)=∏_p

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