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with
$1\leqslant x_{i},y_{i}\leqslant X\;(1\leqslant i\leqslant s)$
. By exploiting sharp estimates for an auxiliary mean value, we obtain bounds for
$I_{s,k,r}(X)$
for
$1\leqslant r\leqslant k-1$
. In particular, when
$s,k\in \mathbb{N}$
satisfy
$k\geqslant 3$
and
$1\leqslant s\leqslant (k^{2}-1)/2$
, we establish the essentially diagonal behaviour
$I_{s,k,1}(X)\ll X^{s+\unicode[STIX]{x1D700}}$
.
We apply multigrade efficient congruencing to estimate Vinogradov’s integral of degree
$k$
for moments of order
$2s$
, establishing strongly diagonal behaviour for
$1\leqslant s\leqslant \frac{1}{2}k(k+1)-\frac{1}{3}k+o(k)$
. In particular, as
$k\rightarrow \infty$
, we confirm the main conjecture in Vinogradov’s mean value theorem for a proportion asymptotically approaching
$100\%$
of the critical interval
$1\leqslant s\leqslant \frac{1}{2}k(k+1)$
.
We investigate sums of mixed powers involving two squares and three biquadrates. In particular, subject to the truth of the Generalised Riemann Hypothesis and the Elliott–Halberstam conjecture, we show that all large natural numbers
$n$
with
$8\nmid n$
,
$n\not\equiv 2~(\text{mod} ~3)$
and
$n\not\equiv 14~(\text{mod} ~16)$
are the sum of two squares and three biquadrates.
We investigate the number of integral solutions possessed by a pair of diagonal cubic equations in a large box. Provided that the number of variables in the system is at least fourteen, and in addition the number of variables in any non-trivial linear combination of the underlying forms is at least eight, we obtain an asymptotic formula for the number of integral solutions consistent with the product of local densities associated with the system.
We 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.
We apply a method of Davenport to improve several estimates for slim exceptional sets associated with the asymptotic formula in Waring’s problem. In particular, we show that the anticipated asymptotic formula in Waring’s problem for sums of seven cubes holds for all but O(N1/3+ε) of the natural numbers not exceeding N.
When p is a prime number, and k1,…,kt are natural numbers with 1≤k1<k2<⋯<kt<p, we show that the simultaneous congruences ∑ t1xkji≡∑ t1ykjimodp (1≤j≤t) possess at most k1⋯ktpt solutions with 1≤xi,yi≤p (1≤i≤t). Analogous conclusions are provided when one or more of the exponents ki is negative.
Consider a system of diagonal equations \begin{equation}\sum_{j=1}^sa_{ij}x_j^k=0\quad (1\le i\le r),\end{equation} satisfying the property that the (fixed) integral coefficient matrix $(a_{ij})$ contains no singular $r\times r$ submatrix. A recent paper of the authors [3] establishes that whenever $k\ge 3$ and $s>(3r+1)2^{k-2}$, then the expected asymptotic formula holds for the number $N(P)$ of integral solutions ${\bf x}$ of ($1{\cdot}1$) with $|x_i|\le P$$(1\le i\le s)$.
An 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.
By avoiding a conventional application of Bessel’s inequality in favour of explicitly
controlling an exponential sum over the exceptional set, in our previous work [9, 10],
we have exploited additional variables so as to enhance exceptional set estimates in
various additive problems of Waring type. In this paper we turn to the problem
of establishing the expected asymptotic formula in Waring’s problem. The methods
introduced herein, although discussed in the context of the asymptotic formula,
should nonetheless provide a useful model for future excursions involving exceptional
sets in additive problems.
This paper concerns systems of r homogeneous diagonal equations of degree k in s variables, with integer
coefficients. Subject to a suitable non-singularity condition, it is shown that the expected asymptotic
formula holds for the number of such systems inside a box [−P,P]s,
provided only that s > (3r+1)2k−2.
By way of comparison, classical methods based on the use
of Hua's lemma would establish a similar
conclusion, provided instead that s > r2k.
We investigate exceptional sets associated with various additive problems involving sums of cubes. By developing a method wherein an exponential sum over the set of exceptions is employed explicitly within the Hardy-Littlewood method, we are better able to exploit excess variables. By way of illustration, we show that the number of odd integers not divisible by 9, and not exceeding
$X$
, that fail to have a representation as the sum of 7 cubes of prime numbers, is
$O\left( {{X}^{23/36+\varepsilon }} \right)$
. For sums of eight cubes of prime numbers, the corresponding number of exceptional integers is
$O\left( {{X}^{11/36+\varepsilon }} \right)$
.
We demonstrate that a pair of additive quintic equations in at least 34 variables has a nontrivial integral solution, subject only to an 11-adic solubility hypothesis. This is achieved by an application of the Hardy–Littlewood method, for which we require a sharp estimate for a 33.998th moment of quintic exponential sums. We are able to employ p-adic iteration in a form that allows the estimation of such a mean value over a complete unit square, thereby providing an approach that is technically simpler than those of previous workers and flexible enough to be applied to related problems.
Given that available technology permits one to establish that almost all natural numbers satisfying appropriate congruence conditions are represented as the sum of three squares of prime numbers, one expects strong estimates to be attainable for exceptional sets in the analogous problem involving sums of four squares of primes. Let E(N)$ denote the number of positive integers not exceeding N that are congruent to 4 modulo 24, yet cannot be written as the sum of four squares of prime numbers. A method is described that shows that for each positive number $\epsilon$, one has $E(N) \ll N^{13/30 + \epsilon}$, thereby exploiting effectively the 'excess' fourth square of a prime so as to improve the recent bound $E(N) \ll N^{13/15 + \epsilon}$ due to J. Liu and M.-C. Liu. It transpires that the ideas underlying this progress permit estimates for exceptional sets in a variety of additive problems to be significantly slimmed whenever sufficiently many excess variables are available. Such ideas are illustrated for several additional problems involving sums of four squares.
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.
It is shown that every sufficiently large integer
congruent to $14$ modulo $240$ may be written as
the sum of $14$ fourth powers of prime numbers, and
that every sufficiently large odd integer may be
written as the sum of $21$ fifth powers of prime
numbers. The respective implicit bounds $14$ and $21$
improve on the previous bounds $15$ (following from
work of Davenport) and $23$ (due to Thanigasalam).
These conclusions are established through the
medium of the Hardy-Littlewood method, the proofs
being somewhat novel in their use of estimates
stemming directly from exponential sums over prime
numbers in combination with the linear sieve, rather
than the conventional methods which `waste' a variable
or two by throwing minor arc estimates down to an
auxiliary mean value estimate based on variables not
restricted to be prime numbers. In the work on fifth
powers, a switching principle is applied to a cognate
problem involving almost primes in order to obtain the
desired conclusion involving prime numbers alone. 2000 Mathematics Subject Classification:
11P05, 11N36, 11L15, 11P55.
Non-trivial estimates for fractional moments of
smooth cubic Weyl sums are developed.
Complemented by bounds for such sums of use on
both the major and minor arcs in a Hardy--Littlewood
dissection, these estimates are applied to derive an
upper bound for the $s$th moment of the smooth cubic
Weyl sum of the expected order of magnitude as soon
as $s\ge 7.691$.
Related arguments demonstrate that all large integers
$n$ are represented as the sum of eight cubes of
natural numbers, all of whose prime divisors are at
most $\exp (c(\log n\log \log n)^{1/2})$, for a
suitable positive number $c$. This conclusion
improves a previous result of G. Harcos in which
nine cubes are required. 1991 Mathematics Subject Classification:
11P05, 11L15, 11P55.
The set of integers represented as the sum of three cubes of natural numbers is widely expected to have positive density (see Hooley [7] for a discussion of this topic). Over the past six decades or so, the pursuit of an acceptable approximation to the latter statement has spawned much of the progress achieved in the theory of the Hardy-Littlewood method, so far as its application to Waring's problem for smaller exponents is concerned. Write R(N) for the number of positive integers not exceeding N which are the sum of three cubes of natural numbers.
Let
$\mathcal{S}$
denote the set of integers representable as a sum of two squares. Since
$\mathcal{S}$
can be described as the unsifted elements of a sieving process of positive dimension, it is to be expected that
$\mathcal{S}$
has many properties in common with the set of prime numbers. In this paper we exhibit “unexpected irregularities” in the distribution of sums of two squares in short intervals, a phenomenon analogous to that discovered by Maier, over a decade ago, in the distribution of prime numbers. To be precise, we show that there are infinitely many short intervals containing considerably more elements of
$\mathcal{S}$
than expected, and infinitely many intervals containing considerably fewer than expected.
Additive representations of natural numbers by mixtures of squares, cubes and
biquadrates belong to the class of more interesting special cases which form the
object of attention for testing the general expectation that any sufficiently large
natural number n is representable in the form
formula here
as soon as the reciprocal sum
[sum ]sj=1k−1j
is reasonably large. With the exception of a handful of very special problems, in the current
state of knowledge the latter reciprocal sum must exceed 2, at the very least,
in order that it be feasible to successfully
apply the Hardy–Littlewood method to treat the corresponding additive problem.
Here we remove a case from the list of those combinations of exponents which have
defied treatment thus far.