For a suitable arithmetic function F and polynomials Q 1 , . . . , Q k in Z[X], Nair and Tenenbaum obtained an upper bound on the short sum x

Abstract. We show that the equation λ 1 n 2 1 + · · · + λ s n 2 s = 0 admits non-trivial solutions in any subset of [N ] of density (log N ) −cs , provided that s 7 and the coefficients λ i ∈ Z {0} sum to zero and satisfy certain sign conditions. This improves upon previous known density bounds of the form (log log N ) −c .

We show that subsets of $\mathbb{R}^n$ of large enough Hausdorff and Fourier dimension contain polynomial patterns of the form \begin{align*} ( x ,\, x + A_1 y ,\, \dots,\, x + A_{k-1} y ,\, x + A_k y + Q(y) e_n ), \quad x \in \mathbb{R}^n,\ y \in \mathbb{R}^m, \end{align*} where $A_i$ are real $n \times m$ matrices, $Q$ is a real polynomial in $m$ variables and $e_n = (0,\dots,0,1)$.Comment: 38 pages, small changes made in light of comments from the referee

We study additive equations of the form s i=1 λiP(ni) = 0 in variables ni ∈ Z d , where the λi are nonzero integers summing up to zero and P = (P1, . . . , Pr) is a system of homogeneous polynomials making the equation translation-invariant. We investigate the solvability of this equation in subsets of density (log N ) −c(P,λ) of a large box [N ] d via the energy-increment method. We obtain positive results for roughly the number of variables currently needed to derive a count of solutions in the complete box [N ] d for the multidimensional systems of large degree studied by Parsell, Prendiville and Wooley. Appealing to estimates from the decoupling theory of Bourgain, Demeter and Guth, we also treat the cases of the monomial curve P = (x, . . . , x k ) and the parabola P = (x, |x| 2 ) for a number of variables close to or equal to the limit of the circle method.

In this note we wish to correct several mistakes which appeared in our paper [1]. Fortunately, all the results claimed in that reference may be recovered, with small modifications to the statements and the argument. We are grateful to Nathan Ng for drawing our attention to an issue in the application of Theorem 6 to divisor sums, and to Régis de la Bretèche for helpful discussions on the problem addressed here.Congruence conditions. We use the notation as well as page, equation and theorem numbers from [1]. The most problematic part of our paper is the lower bound in Theorem 6, which is incorrect as stated, and as a consequence the upper bounds in Theorem 5 and Corollaries 1-2 are not sharp as claimed (although they are still valid). We explain here how to recover a lower bound on the sum considered in Theorem 6, together with a matching upper bound in Theorems 5 and 6, by modifying the function ρ R (a 1 , . . . , a r ) which encodes certain polynomial congruences.The source of the error in Theorem 6 is at the end of p. 423, where the last displayed equation is wrong: the conditionsε 3 alone are insufficient to guarantee that the a i are the unique integers such that, but we may well have p| ji a j , preventing us from concluding that p > x ε 3 . To address this we introduce the following property, dependent on integers a 1 , . . . , a r , n:The workaround we find is to replace the function ρ R defined in (2·8), in all instances where it appears, by the functionρ R defined bywe explain shortly why this is possible. First observe that, by the Chinese remainder theorem,ρ R is also multiplicative, and sinceρ R ρ R it also satisfies the upper bound (2·9). Thereforeρ R behaves as ρ R for analytic purposes, and it remains to show that the substitution can be performed. In our paper, the function ρ R arises exclusively through applications available at https://www.cambridge.org/core/terms. https://doi