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We prove the following theorem: for all positive integers $b$ there exists a positive integer $k$, such that for every finite set $A$ of integers with cardinality $|A| > 1$, we have either $$ |A + ... + A| \geq |A|^b$$ or $$ |A \cdot ... \cdot A| \geq |A|^b$$ where $A + ... + A$ and $A \cdot ... \cdot A$ are the collections of $k$-fold sums and products of elements of $A$ respectively. This is progress towards a conjecture of Erd��s and Szemer��di on sum and product sets., 33 pages, no figures, submitted, J. Amer. Math. Soc. (Proxy submission)

We investigate the moduli spaces of stable rank two reflexive sheaves on P3 with small Chern classes. As an application to curves of low degree in P3, we prove the curve has maximal rank and that the corresponding Hilbert scheme is irreducible and unirational. Introduction. In the past few years, the subject of vector bundles on projective spaces and, in particular, the case of rank 2 on P3, has received much attention. Several basic theorems have been proved, e.g., the existence of the moduli space of stable vector bundles [M], though so far very few of these moduli spaces have been studied in detail. The interest of vector bundles partly stems from their connection with curves. However, the class of curves obtained in this way is rather restricted. Recently, Hartshorne [SRS] has focused attention on reflexive sheaves of rank 2 on P3. On the one hand, most results proved for vector bundles also turn out to be true for reflexive sheaves. On the other hand, reflexive sheaves have two significant advantages. First, they correspond to quite general curves; second, and most importantly, one has the "reduction step" introduced by Hartshorne, which is an effective tool in studying vector bundles, by relating them to simpler reflexive sheaves. In this paper, we investigate the moduli spaces of stable rank 2 reflexive sheaves on P3 with c2 3. For c2 < 2, we prove the moduli spaces are irreducible, nonsingular and rational; we also classify the related unstable planes. For c2= 3, in most cases we show that the moduli space is irreducible and unirational. As one of the applications to curves of low degree in P3 (cf. [SRS, 4.1]), we prove that the curve has maximal rank and that the corresponding Hilbert scheme is irreducible and unirational. Hence, as a corollary, we conclude that the variety of moduli '1g of curves of genus g is unirational, for g = 5, 6, 7, 8. In ?1, we give facts on multiple lines and deduce a criterion for unstable planes. ?2 contains our classification of semistable sheaves with c2 < 2. ?3 is about sheaves with c2 = 3. ?4 gives the applications to curves. ACKNOWLEDGMENT. I would like to thank my thesis advisor Robin Hartshorne. He has been very generous in his advice and time. In addition, he attracted to Received by the editors September 1, 1982 and, in revised form, March 18, 1983. 1980 Mathematics Subject Classification. Primary 14F05; Secondary 14D22. I1984 American Mathematical Society 0002-9947/84 $1.00 + $.25 per page

We present lower bounds for the orbit length of reduction modulo primes of parametric polynomial dynamical systems defined over the integers, under a suitable hypothesis on its set of preperiodic points over C \mathbb {C} . Applying recent results of Baker and DeMarco (2011) and of Ghioca, Krieger, Nguyen and Ye (2017), we obtain explicit families of parametric polynomials and initial points such that the reductions modulo primes have long orbits, for all but a finite number of values of the parameters. This generalizes a previous lower bound due to Chang (2015). As a by-product, we also slightly improve a result of Silverman (2008) and recover a result of Akbary and Ghioca (2009) as special extreme cases of our estimates.