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Abstract Let satisfy . Freĭman's theorem shows that when , there exists such that all large integers are represented in the form , with , if and only if diverges. We make this theorem effective by showing that, for each fixed , it suffices to impose the conditionMore is established when the sequence of exponents forms an arithmetic progression. Thus, for example, when and , all large integers are represented in the form , with .
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Abstract When$$k\geqslant 4$$and$$0\leqslant d\leqslant (k-2)/4$$, we consider the system of Diophantine equations\begin{align*}x_1^j+\ldots +x_k^j=y_1^j+\ldots +y_k^j\quad (1\leqslant j\leqslant k,\, j\ne k-d).\end{align*}We show that in this cousin of a Vinogradov system, there is a paucity of non-diagonal positive integral solutions. Our quantitative estimates are particularly sharp when$$d=o\!\left(k^{1/4}\right)$$.more » « less
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Abstract Let$$\varphi _1,\ldots ,\varphi _r\in {\mathbb Z}[z_1,\ldots z_k]$$be integral linear combinations of elementary symmetric polynomials with$$\text {deg}(\varphi _j)=k_j\ (1\le j\le r)$$, where$$1\le k_1<\cdots . Subject to the condition$$k_1+\cdots +k_r\ge \tfrac {1}{2}k(k-~1)+2$$, we show that there is a paucity of nondiagonal solutions to the Diophantine system$$\varphi _j({\mathbf x})=\varphi _j({\mathbf y})\ (1\le j\le r)$$.more » « less
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Abstract Let {G(k)}denote the least numbershaving the property that everysufficiently large natural number is the sum of at mostspositive integralk-th powers.Then for all {k\in\mathbb{N}}, one has G(k)\leqslant\lceil k(\log k+4.20032)\rceil. Our new methods improve on all bounds available hitherto when {k\geqslant 14}.more » « less
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Abstract When k and s are natural numbers and $${\mathbf h}\in {\mathbb Z}^k$$, denote by $$J_{s,k}(X;\,{\mathbf h})$$ the number of integral solutions of the system $$ \sum_{i=1}^s(x_i^j-y_i^j)=h_j\quad (1\leqslant j\leqslant k), $$ with $$1\leqslant x_i,y_i\leqslant X$$. When $$s\lt k(k+1)/2$$ and $$(h_1,\ldots ,h_{k-1})\ne {\mathbf 0}$$, Brandes and Hughes have shown that $$J_{s,k}(X;\,{\mathbf h})=o(X^s)$$. In this paper we improve on quantitative aspects of this result, and, subject to an extension of the main conjecture in Vinogradov’s mean value theorem, we obtain an asymptotic formula for $$J_{s,k}(X;\,{\mathbf h})$$ in the critical case $s=k(k+1)/2$. The latter requires minor arc estimates going beyond square-root cancellation.more » « less
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Under appropriate local solubility conditions on $$\bfn$$, we obtain an asymptotic formula for $$A_{s,k}(\bfn)$$ when $$s\ge k(k+1)$$. This establishes a local-global principle in the Hilbert-Kamke problem at the convexity barrier. Our arguments involve minor arc estimates going beyond square-root cancellation.more » « lessFree, publicly-accessible full text available September 1, 2026
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We prove a function field analog of Weyl's classical theorem on equidistribution of polynomial sequences. Our result covers the case in which the degree of the polynomial is greater than or equal to the characteristic of the field, which is a natural barrier when applying the Weyl differencing process to function fields. We also discuss applications to van der Corput, intersective and Glasner sets in function fields.more » « lessFree, publicly-accessible full text available July 9, 2026
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We investigate k-superirreducible polynomials, by which we mean irreducible polynomials that remain irreducible under any polynomial substitution of positive degree at most k. Let F be a finite field of characteristic p. We show that no 2-superirreducible polynomials exist in F[t] when p=2 and that no such polynomials of odd degree exist when p is odd. We address the remaining case in which p is odd and the polynomials have even degree by giving an explicit formula for the number of monic 2-superirreducible polynomials having even degree d. This formula is analogous to that given by Gauss for the number of monic irreducible polynomials of given degree over a finite field. We discuss the associated asymptotic behaviour when either the degree of the polynomial or the size of the finite field tends to infinity.more » « lessFree, publicly-accessible full text available May 1, 2026
