Estimate of a Trigonometrical Sum Involving Naturals with Binary Decompositions of a Special Kind

TL;DR

This paper estimates a specific trigonometric sum over natural numbers with binary decompositions having an even number of 1s, advancing understanding of such sums in number theory.

Contribution

It introduces a novel estimation method for sums involving natural numbers with binary decomposition constraints, focusing on quadratic exponential sums.

Findings

Derived bounds for the sum involving binary-decomposed natural numbers.

Extended classical exponential sum estimates to a specialized subset of naturals.

Provided analytical techniques applicable to similar binary-restricted sum problems.

Abstract

Let $\mathbb{N}_0$ be a class of natural numbers whose binary decompositions has even number of 1. We estimate of the sum $\sum\limits_{n\in \mathbf{N}_0,n\le X}\exp(2\pi i \alpha n^2)$.