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Waring's problem with almost proportional summands

Zarullo Rakhmonov, Firuz Rakhmonov

Abstract

For $n \geq 3$, an asymptotic formula is derived for the number of representations of a sufficiently large natural number $N$ as a sum of $r = 2^n + 1$ summands, each of which is an $n$-th power of natural numbers $x_i$, $i = \overline{1, r}$, satisfying the conditions $$ |x_i^n-μ_iN|\le H,\qquad H\ge N^{1-θ(n,r)+\varepsilon},\qquad θ(n,r)=\frac2{(r+1)(n^2-n)}, $$ where $μ_1, \ldots, μ_r$ are positive fixed numbers, and $μ_1 + \ldots + μ_n = 1$. This result strengthens the theorem of E.M.Wright.

Waring's problem with almost proportional summands

Abstract

For , an asymptotic formula is derived for the number of representations of a sufficiently large natural number as a sum of summands, each of which is an -th power of natural numbers , , satisfying the conditions where are positive fixed numbers, and . This result strengthens the theorem of E.M.Wright.

Paper Structure

This paper contains 12 sections, 15 theorems, 197 equations.

Table of Contents

  1. Introduction
  2. Well-known lemmas
  3. Proof of Theorem \ref{['Teor ob poved kor trig summi Weyl']} on the behavior of short Weyl exponential sums in major arcs
  4. Proof of Theorem \ref{['Teorema-obobsh-Hua']} on the average value of the short exponential sum $T(\alpha;x,y)$
  5. Proof of Theorem \ref{['TeorAsForEstKubPr']}
  6. Calculation of the integral $J(\mathfrak{M}_1)$
  7. Estimation of $R_2(\mathfrak{M}_1)$
  8. Estimation of $R_1(\mathfrak{M}_1)$
  9. Calculation of the integral $\mathscr{A}(r,q)$
  10. Derivation of the Asymptotic Formula for the Integral $J(\mathfrak{M}_1)$
  11. Estimation of the integral $J(\mathfrak{M}_2)$
  12. Estimation of the integral $J(\mathfrak{m})$

Key Result

Theorem 1.1

Let $N$ be a sufficiently large natural number, $n\ge3$ be a natural number, $r=2^n+1$, and $\mu_1,\ldots,\mu_r$ be positive fixed numbers satisfying the condition $J_{n,r}(N,H)$ is the number of solutions to the Diophantine equation (Formula-x1n+...+xrn=N) under the conditions Then, for $H\ge N^{1-\theta(n,r)+\varepsilon}$, the following asymptotic formula holds: where $\gamma(n,r)$ is an abso

Theorems & Definitions (17)

  • Theorem 1.1
  • Corollary 1.1
  • Theorem 1.2
  • Corollary 1.2
  • Corollary 1.3
  • Theorem 1.3
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • ...and 7 more