Small fractional parts of additive forms

We show how the methods of Vaughan & Wooley, which have proved fruitful in dealing with Waring’s problem, may also be used to investigate the fractional parts of an additive form. Results are obtained which are near to best possible for forms with algebraic coefficients. New results are also obtained in the general case. Extensions are given to make several additive forms simultaneously small. The key ingredients in this work are: mean value theorems for exponential sums, the use of a small common factor for the integer variables, and the large sieve inequality.

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Arithmetic of higher-dimensional orbifolds and a mixed Waring problem

Mathematische Zeitschrift ◽

10.1007/s00209-021-02695-w ◽

2021 ◽

Author(s):

Tim Browning ◽

Shuntaro Yamagishi

Keyword(s):

Circle Method ◽

Rational Points ◽

Mean Value ◽

Mean Value Theorem ◽

Waring’S Problem ◽

Waring Problem ◽

Waring's Problem ◽

Main Conjecture ◽

Higher Dimensional ◽

Thin Set

AbstractWe study the density of rational points on a higher-dimensional orbifold $$(\mathbb {P}^{n-1},\Delta )$$ ( P n - 1 , Δ ) when $$\Delta $$ Δ is a $$\mathbb {Q}$$ Q -divisor involving hyperplanes. This allows us to address a question of Tanimoto about whether the set of rational points on such an orbifold constitutes a thin set. Our approach relies on the Hardy–Littlewood circle method to first study an asymptotic version of Waring’s problem for mixed powers. In doing so we make crucial use of the recent resolution of the main conjecture in Vinogradov’s mean value theorem, due to Bourgain–Demeter–Guth and Wooley.

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On Waring’s problem: Three cubes and a sixth power

Nagoya Mathematical Journal ◽

10.1017/s0027763000007893 ◽

2001 ◽

Vol 163 ◽

pp. 13-53 ◽

Cited By ~ 4

Author(s):

Jörg Brüdern ◽

Trevor D. Wooley

Keyword(s):

Exponential Sums ◽

Large Integer ◽

Natural Numbers ◽

Waring’S Problem ◽

Smooth Numbers ◽

Waring's Problem ◽

Order Of Magnitude ◽

Almost All ◽

Hardy Littlewood Method

We establish that almost all natural numbers not congruent to 5 modulo 9 are the sum of three cubes and a sixth power of natural numbers, and show, moreover, that the number of such representations is almost always of the expected order of magnitude. As a corollary, the number of representations of a large integer as the sum of six cubes and two sixth powers has the expected order of magnitude. Our results depend on a certain seventh moment of cubic Weyl sums restricted to minor arcs, the latest developments in the theory of exponential sums over smooth numbers, and recent technology for controlling the major arcs in the Hardy-Littlewood method, together with the use of a novel quasi-smooth set of integers.

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Weyl sums, mean value estimates, and Waring's problem with friable numbers

Acta Arithmetica ◽

10.4064/aa8448-7-2016 ◽

2016 ◽

Vol 176 (3) ◽

pp. 249-299 ◽

Cited By ~ 1

Author(s):

Sary Drappeau ◽

Xuancheng Shao

Keyword(s):

Mean Value ◽

Waring’S Problem ◽

Waring's Problem

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A NEW BOUND FOR THE LARGE SIEVE INEQUALITY WITH POWER MODULI

International Journal of Number Theory ◽

10.1142/s179304211250039x ◽

2012 ◽

Vol 08 (03) ◽

pp. 689-695 ◽

Cited By ~ 5

Author(s):

KARIN HALUPCZOK

Keyword(s):

Mean Value ◽

Mean Value Theorem ◽

Large Sieve ◽

Large Sieve Inequality

We give a new bound for the large sieve inequality with power moduli qk that is uniform in k. The proof uses a new theorem due to Wooley from his work [Vinogradov's mean value theorem via efficient congruencing, to appear in Ann. of Math.] on efficient congruencing.

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On Waring’s problem: Three cubes and a minicube

Nagoya Mathematical Journal ◽

10.1215/00277630-2010-012 ◽

2010 ◽

Vol 200 ◽

pp. 59-91 ◽

Cited By ~ 1

Author(s):

Jörg Brüdern ◽

Trevor D. Wooley

Keyword(s):

Exponential Sums ◽

Independent Interest ◽

Natural Numbers ◽

Waring’S Problem ◽

Waring's Problem ◽

Almost All ◽

Positive Integers

AbstractWe establish that almost all natural numbers n are the sum of four cubes of positive integers, one of which is no larger than n5/36. The proof makes use of an estimate for a certain eighth moment of cubic exponential sums, restricted to minor arcs only, of independent interest.

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Further improvements in Waring’s problem. Ill Eighth powers

Philosophical Transactions of the Royal Society of London Series A Physical and Engineering Sciences ◽

10.1098/rsta.1993.0137 ◽

1993 ◽

Vol 345 (1676) ◽

pp. 385-396 ◽

Cited By ~ 4

Keyword(s):

Diophantine Equation ◽

Exponential Sums ◽

Natural Numbers ◽

Prime Divisors ◽

Waring’S Problem ◽

Waring's Problem

In this paper we continue our development of the methods of Vaughan & Wooley, these being based on the use of exponential sums over integers having only small prime divisors. On this occasion we concentrate on improvements in the estimation of the contribution of the major arcs arising in the efficient differencing process. By considering the underlying diophantine equation, we are able to replace certain smooth Weyl sums by classical Weyl sums, and thus we are able to utilize a number of pruning processes to facilitate our analysis. These methods lead to improvements in Waring’s problem for larger k . In this instance we prove that G (8) ⩽ 42, which is to say that all sufficiently large natural numbers are the sum of at most 42 eighth powers of integers. This improves on the earlier bound G (8) ⩽ 43.

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On Waring’s problem: Three cubes and a minicube

Nagoya Mathematical Journal ◽

10.1017/s0027763000010175 ◽

2010 ◽

Vol 200 ◽

pp. 59-91

Author(s):

Jörg Brüdern ◽

Trevor D. Wooley

Keyword(s):

Exponential Sums ◽

Independent Interest ◽

Natural Numbers ◽

Waring’S Problem ◽

Waring's Problem ◽

Almost All ◽

Positive Integers

AbstractWe establish that almost all natural numbersnare the sum of four cubes of positive integers, one of which is no larger thann5/36. The proof makes use of an estimate for a certain eighth moment of cubic exponential sums, restricted to minor arcs only, of independent interest.

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