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Propositions 1.1–1.3 stated below contribute to results and certain problems considered in [E-S], on the behavior of products $$\Pi^n_1(1-z^{a_j}),1\leq{a_1}...\leq{a_n}$$ integers. In the discussion below, {a1,..., an} will be either a proportional subset of {1,..., n} or a set of large arithmetic diameter.

We obtain smoothing estimates for certain nonlinear convolution operators on prime fields, leading to quantitative nonlinear Roth type theorems. Compared with the usual linear setting (i.e. arithmetic progressions), the nonlinear nature of the operators leads to different phenomena, both qualitatively and quantitatively.

We use bounds of mixed character sums modulo a square-free integer q of a special structure to estimate the density of integer points on the hypersurface $$\begin{aligned} f_1(x_1) + \cdots + f_n(x_n) =a x_1^{k_1} \ldots x_n^{k_n} \end{aligned}$$ for some polynomials $$f_i \in {\mathbb {Z}}[X]$$ and nonzero integers a and $$k_i$$ , $$i=1, \ldots , n$$ . In the case of $$\begin{aligned} f_1(X) = \cdots = f_n(X) = X^2\quad \text{ and }\quad k_1 = \cdots = k_n =1 \end{aligned}$$ the above hypersurface is known as the Markoff–Hurwitz hypersurface, while for $$\begin{aligned} f_1(X) = \cdots = f_n(X) = X^n\quad \text{ and }\quad k_1 = \cdots = k_n =1 \end{aligned}$$ it is known as the Dwork hypersurface. Our results are substantially stronger than those known for general hypersurfaces.

In this note, we use ‘classical’ methods to obtain sum-product theorems for subsets A⊂$$ \mathbb{F} $$ p .

In this paper we extend the exponential sum results from [BK] and [BGK] for prime moduli to composite moduli q involving a bounded number of prime factors. In particular, we obtain nontrivial bounds on the exponential sums associated to multiplicative subgroups H of size qδ, for any given δ > 0. The method consists in first establishing a ‘sumproduct theorem’ for general subsets A of $$\mathbb{Z}^{q} $$ . If q is prime, the statement, proven in [BKT], expresses simply that either the sum-set A + A or the product-set A.A is significantly larger than A, unless |A| is near q. For composite q, the presence of nontrivial subrings requires a more complicated dichotomy, which is established here. With this sum-product theorem at hand, the methods from [BGK] may then be adapted to the present context with composite moduli. They rely essentially on harmonic analysis and graph-theoretical results such as Gowers’ quantitative version of the Balog–Szemeredi theorem. As a corollary, we get nontrivial bounds for the ‘Heilbronn-type’ exponential sums when q = p r (p prime) for all r. Only the case r = 2 has been treated earlier in works of Heath-Brown and Heath-Brown and Konyagin (using Stepanov’s method). We also get exponential sum estimates for (possibly incomplete) sums involving exponential functions, as considered for instance in [KS].

LetA be a finite subset of some normed division algebra over ℝ with cardinality ⋎A⋎. We prove that either the sum set or the product set ofA has cardinality ⋎A⋎1+δ for some δ>0.

Let X⊂ℙN be either a threefold of Calabi–Yau or of general type (embedded with r KX). In this article we give lower and upper bounds, linear on the degree of X and N, for the Euler number of X. As a corollary we obtain the boundedness of the region described by the Chern ratios \(\) of threefolds with ample canonical bundle and a new upper bound for the number of nodes of a complete intersection threefold.

We bound the third Chern number of a minimal smooth threefold with ample canonical bundle by quadratic functions in the first two Chern numbers.

Let E be a stable rank 2 vector bundle on n )3. Assume E is normalized, i.e., c~(E)=0 or 1 . By Grothedieck's theorem the restriction of E on a line L is isomorphic to (~L(bL)O(gL(--bL) if Cl=O[(gL(bL--1)t~OL(--b L) if C1=-1 ] , for some nonnegative integer b L. This b L is an upper semicontinuous function of L, which in char. 0, according to the Grauert-Miilich theorem [G-M], assumes the value 0 on an open set. If bE>0, then L is said to be a jumping line of order b L. For c 1 =0, Barth [B] has shown that the set of jumping lines can be given a scheme structure, making it a divisor S E on the Grassmannian G=G(1, 3) which via G ~IP 5, has degree 2c2(E), and for any line L in IP 3, b L < PL(SE), where #~.(S~) is the multiplicity of S t at L. Since any point on a projective variety of degree k has multiplicity < k, it follows that b L < 2c 2. However, this bound is not sharp. In this paper, we will give a sharp bound for b L, valid for arbitrary c r Namely, if c~ =0