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Waring's problem for pseudo-polynomials

Manfred G. Madritsch

TL;DR

This work extends Waring’s problem to sums of floor values of a pseudo-polynomial, proving an asymptotic count for representations $r_{f,s}(N)$ with leading term governed by a Gamma-weighted power of $N$ when $s$ exceeds a explicit threshold. The authors adapt the circle method, decompose the unit interval into major and minor arcs, and compute the main term via a singular integral that yields Gamma-function constants, while bounding the minor-arc contribution through refined exponential-sum estimates anchored in Vinogradov's mean value theorem. Key innovations include handling non-integer exponents in the pseudo-polynomial, controlling the floor-function with Fourier-analytic tools, and extending prior results that required $c>12$ to the broader range $c>1$. The results significantly advance Waring-type representations for non-classical summands and provide a rigorous asymptotic formula with an explicit error exponent. The methods integrate classical circle-method techniques with modern mean-value bounds to accommodate the fractional exponents appearing in $f$.

Abstract

Waring's problem has a long history in additive number theory. In its original form it deals with the representability of every positive integer as sum of $k$-th powers with integer $k$. Instead of these powers we deal with pseudo-polynomials in this paper. A pseudo-polynomial is a ``polynomial'' with at least one exponent not being an integer. Our work extends earlier results on the related problem of Waring for arbitrary real powers $k>12$ by Deshouillers and Arkhipov and Zhitkov.

Waring's problem for pseudo-polynomials

TL;DR

This work extends Waring’s problem to sums of floor values of a pseudo-polynomial, proving an asymptotic count for representations with leading term governed by a Gamma-weighted power of when exceeds a explicit threshold. The authors adapt the circle method, decompose the unit interval into major and minor arcs, and compute the main term via a singular integral that yields Gamma-function constants, while bounding the minor-arc contribution through refined exponential-sum estimates anchored in Vinogradov's mean value theorem. Key innovations include handling non-integer exponents in the pseudo-polynomial, controlling the floor-function with Fourier-analytic tools, and extending prior results that required to the broader range . The results significantly advance Waring-type representations for non-classical summands and provide a rigorous asymptotic formula with an explicit error exponent. The methods integrate classical circle-method techniques with modern mean-value bounds to accommodate the fractional exponents appearing in .

Abstract

Waring's problem has a long history in additive number theory. In its original form it deals with the representability of every positive integer as sum of -th powers with integer . Instead of these powers we deal with pseudo-polynomials in this paper. A pseudo-polynomial is a ``polynomial'' with at least one exponent not being an integer. Our work extends earlier results on the related problem of Waring for arbitrary real powers by Deshouillers and Arkhipov and Zhitkov.
Paper Structure (8 sections, 13 theorems, 91 equations)

This paper contains 8 sections, 13 theorems, 91 equations.

Table of Contents

  1. Introduction
  2. The setup
  3. The major arc
  4. The singular integral
  5. The floor function
  6. Exponential sum estimates
  7. The minor arc
  8. Proof of Theorem \ref{['thm:main']}

Key Result

Theorem 1.1

Let $f$ be a pseudo-polynomial as in eq:pseudo-polynomial and set $\rho=\min(\theta_d-\theta_{d-1},\tfrac{1}{6})$ with $\theta_0=0$. Suppose that Then there exists $\delta>0$ such that

Theorems & Definitions (21)

  • Theorem 1.1
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Proposition 3.3
  • proof
  • Lemma 4.1
  • proof
  • Lemma 4.2: nathanson1996:additive_number_theory*Lemma 5.3
  • ...and 11 more