scholarly journals On Waring’s problem: Three cubes and a minicube

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.

scholarly journals On Waring’s problem: Three cubes and a minicube

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 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.

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

2001 ◽  
Vol 163 ◽  
pp. 13-53 ◽  
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.

Further improvements in Waring’s problem. Ill Eighth powers

1993 ◽  
Vol 345 (1676) ◽  
pp. 385-396 ◽  
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.

scholarly journals A light-weight version of Waring's problem

2004 ◽  
Vol 76 (3) ◽  
pp. 303-316 ◽  
Author(s):  
Trevor D. Wooley
Keyword(s):  
Asymptotic Formula ◽  
Large Integer ◽  
Light Weight ◽  
Natural Numbers ◽  
Waring’S Problem ◽  
Waring's Problem ◽  
Large Solutions ◽  
Homogeneous Weight

AbstractAn asymptotic formula is established for the number of representations of a large integer as the sum of kth powers of natural numbers, in which each representation is counted with a homogeneous weight that de-emphasises the large solutions. Such an asymptotic formula necessarily fails when this weight is excessively light.

On Waring's problem for cubes

1991 ◽  
Vol 109 (2) ◽  
pp. 229-256 ◽  
Author(s):  
Jörg Brüdern
Keyword(s):  
Singular Series ◽  
Natural Numbers ◽  
Additive Theory ◽  
Waring’S Problem ◽  
Waring's Problem ◽  
Quantitative Form ◽  
Image Position ◽  
Theory Of Numbers

A classical conjecture in the additive theory of numbers is that all sufficiently large natural numbers may be written as the sum of four positive cubes of integers. This is known as the Four Cubes Problem, and since the pioneering work of Hardy and Littlewood one expects a much more precise quantitative form of the conjecture to hold. Let v(n) be the number of representations of n in the proposed manner. Then the expected formula takes the shapewhere (n) is the singular series associated with four cubes as familiar in the Hardy–Littlewood theory.

scholarly journals COUNTING THE NUMBER OF SOLUTIONS TO THE ERDŐS–STRAUS EQUATION ON UNIT FRACTIONS

2013 ◽  
Vol 94 (1) ◽  
pp. 50-105 ◽  
Author(s):  
CHRISTIAN ELSHOLTZ ◽  
TERENCE TAO
Keyword(s):  
Positive Integer ◽  
Lower Bounds ◽  
Diophantine Equation ◽  
Upper And Lower Bounds ◽  
Natural Numbers ◽  
Number Of Solutions ◽  
Waring’S Problem ◽  
Unit Fractions ◽  
Image Position ◽  
Positive Integers

AbstractFor any positive integer $n$, let $f(n)$ denote the number of solutions to the Diophantine equation $$\begin{eqnarray*}\frac{4}{n} = \frac{1}{x} + \frac{1}{y} + \frac{1}{z}\end{eqnarray*}$$ with $x, y, z$ positive integers. The Erdős–Straus conjecture asserts that $f(n)\gt 0$ for every $n\geq 2$. In this paper we obtain a number of upper and lower bounds for $f(n)$ or $f(p)$ for typical values of natural numbers $n$ and primes $p$. For instance, we establish that $$\begin{eqnarray*}N\hspace{0.167em} {\mathop{\log }\nolimits }^{2} N\ll \displaystyle \sum _{p\leq N}f(p)\ll N\hspace{0.167em} {\mathop{\log }\nolimits }^{2} N\log \log N.\end{eqnarray*}$$ These upper and lower bounds show that a typical prime has a small number of solutions to the Erdős–Straus Diophantine equation; small, when compared with other additive problems, like Waring’s problem.

Note on the representation of integers as sums of certain powers

1969 ◽  
Vol 65 (2) ◽  
pp. 445-446 ◽  
Author(s):  
K. Thanigasalam
Keyword(s):  
Asymptotic Formula ◽  
Natural Number ◽  
Natural Numbers ◽  
Waring’S Problem ◽  
Waring's Problem ◽  
Representation Of Integers ◽  
Image Position ◽  
Large Natural Number

In the paper entitled ‘Asymptotic formula in a generalized Waring's problem’, I established an asymptotic formula for the number of representations of a large natural number N in the formwhere x1, x2, …, x7 and k are natural numbers with k ≥ 4 (see (2) Theorem 2).

Small fractional parts of additive forms

1993 ◽  
Vol 345 (1676) ◽  
pp. 327-338 ◽  
Keyword(s):  
Exponential Sums ◽  
Common Factor ◽  
Mean Value ◽  
Large Sieve ◽  
Waring’S Problem ◽  
Mean Value Theorems ◽  
Integer Variables ◽  
Waring's Problem ◽  
Large Sieve Inequality ◽  
Additive Form

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.

Fractional parts of powers of rationals

1975 ◽  
Vol 77 (2) ◽  
pp. 269-279 ◽  
Author(s):  
A. Baker ◽  
J. Coates
Keyword(s):  
Finite Number ◽  
Waring’S Problem ◽  
Waring's Problem ◽  
Image Position ◽  
Upper Estimate ◽  
Positive Integers

Mahler (5) proved in 1957 that for any rational a/q, where a, q are relatively prime integers with a > q ≥ 2, and any ε > 0, there exist only finitely many positive integers n such that ∥(a/q)n∥ < e−εn; here ∥x∥ denotes the distance of x from the nearest integer taken positively. In particular there exist only finitely many n such thatand, as Mahler observed, this implies that the number g(k) occurring in Waring's problem is given byfor all but a finite number of values of k. It would plainly be of interest to establish a bound for the exceptional k and this would follow from an upper estimate for the integers n for which (1) holds. But Mahler's work was based on Ridout's generalization of Roth's theorem and, as is well known, the latter result is ineffective.

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