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We provide new estimates for smooth Weyl sums on minor arcs and explore their consequences for the distribution of the fractional parts of αnk$\alpha n^k$. In particular, when k⩾6$k\geqslant 6$ and ρ(k)$\rho (k)$ is defined via the relation ρ(k)−1=k(logk+8.02113)$\rho (k)^{-1}=k(\log k+8.02113)$, then for all large numbers N$N$ there is an integer n$n$ with 1⩽n⩽N$1\leqslant n\leqslant N$ for which ∥αnk∥⩽N−ρ(k)$\Vert \alpha n^k\Vert \leqslant N^{-\rho (k)}$. [ABSTRACT FROM AUTHOR]

Let ki∈N$k_i\in {\mathbb {N}}$(i⩾1)$(i\geqslant 1)$ satisfy 2⩽k1⩽k2⩽⋯$2\leqslant k_1\leqslant k_2\leqslant \cdots$. Freĭman's theorem shows that when j∈N$j\in {\mathbb {N}}$, there exists s=s(j)∈N$s=s(j)\in {\mathbb {N}}$ such that all large integers n$n$ are represented in the form n=x1kj+x2kj+1+⋯+xskj+s−1$n=x_1^{k_j}+x_2^{k_{j+1}}+\cdots +x_s^{k_{j+s-1}}$, with xi∈N$x_i\in {\mathbb {N}}$, if and only if ∑ki−1$\sum k_i^{-1}$ diverges. We make this theorem effective by showing that, for each fixed j$j$, it suffices to impose the condition ∑i=j∞ki−1⩾2logkj+4.71.$$\begin{equation*} \sum _{i=j}^\infty k_i^{-1}\geqslant 2\log k_j +4.71. \end{equation*}$$More is established when the sequence of exponents forms an arithmetic progression. Thus, for example, when k∈N$k\in {\mathbb {N}}$ and s⩾100(k+1)2$s\geqslant 100(k+1)^2$, all large integers n$n$ are represented in the form n=x1k+x2k+1+⋯+xsk+s−1$n=x_1^k+x_2^{k+1}+\cdots +x_s^{k+s-1}$, with xi∈N$x_i\in {\mathbb {N}}$. [ABSTRACT FROM AUTHOR]

We consider certain systems of three linked simultaneous diagonal equations in ten variables with total degree exceeding five. By means of a complification argument, we obtain an asymptotic formula for the number of integral solutions of this system of bounded height that resolves the associated paucity problem. [ABSTRACT FROM AUTHOR]

We apply the circle method to obtain an asymptotic formula for the number of integral points on a certain sliced cubic hypersurface related to the Segre cubic. Unusually, the major and minor arc integrals in this application are both positive and of the same order of magnitude. [ABSTRACT FROM AUTHOR]

A strong form of the Manin–Peyre conjecture with a power-saving error term is proved for a certain cubic fourfold. [ABSTRACT FROM PUBLISHER]

Paucity is established for a system of diagonal diophantine equations, in which the degrees are the odd numbers in ascending order. [ABSTRACT FROM PUBLISHER]

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. [ABSTRACT FROM PUBLISHER]

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. [ABSTRACT FROM AUTHOR]

It was shown by Davenport and Roth [7] that the values taken byat integer points ( x1, …, x8) 8 are dense on the real line, providing at least one of the ratios λi/λj, is irrational. Here and throughout, λi denote such nonzero real numbers. More precisely, Liu, Ng and Tsang [8] showed that for all the inequalityhas infinitely many solutions in integers. Later Baker [1] obtained the same result in the enlarged range . In this note we improve this further, the progress being considerable. [ABSTRACT FROM PUBLISHER]

A classical problem in the additive theory of numbers is the determination of the minimal s such that for all sufficiently large n the equationis solvable in natural numbers xk. Improving on earlier results the author [2] has been able to prove that one may take s = 18. In a survey article W. Schwarz asked for an analogue for diophantine inequalities [6]. As a first contribution to this subject we proveTheorem. Let λ2, …, λ23 be nonzero real numbers, λ2/λ3 irrational. Then the values taken byat integer points ( x1, …, x22) are dense on the real line. [ABSTRACT FROM PUBLISHER]