The trigonometric polynomial on sums of two squares, an additive problem and generalisation

2025/09/27 by Ramare, Olivier, Viswanadham, GK · 1 citation
#11L07 (Primary) 11M26 #11N25 #11P55 (Secondary) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2509.23260

Abstract

Let B be the set of odd integers that are sums of two coprime squares. We prove that the trigonometric polynomial S(α;N)=∑_{b∈ B,b≤ N} e(bα) satisfies (S(α; N))/(N/√(log N))lt;lt;_A,A' (1)/(ϕ(q)) + √((q)/(N))(log N)⁷ +(1)/((log N)^A) for any A,A'≥ 0 and when (a,q)=1 and |qα-a|≤ (log N)^A'/N. We use this estimate together with a variant of the circle method influenced by Green and Tao's Transference Principle to obtain the number of representations of a large enough odd integer N as a sum b+b₁+b₂, where b∈ B while b₁ (resp. b₂) belongs to a general subset B₁ (resp. B₂) of B of relative positive density. We further show that the above bound is effective when 0≤ A<1/2.

Citations

• Notes on restriction theory in the primes
• Representation of integers by cyclotomic binary forms
• An L-function free proof of Hua's Theorem on sums of five prime squares
• Linear Equations in Primes
• Restriction theory of the Selberg sieve, with applications
• The primes contain arbitrarily long arithmetic progressions
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Cited by

• Restriction estimates with sifted integers

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