Building on recent work of Masser concerning algebraic values of the Riemann zeta function, we prove two general results about the scarcity of algebraic points on the graphs of certain restrictions of certain analytic functions. For any of the graphs to which our results apply and any positive integer d, we show that there are at most C (log H) 3+ε algebraic points of degree at most d and multiplicative height at most H on that graph. In particular, we obtain this conclusion for any restriction of Γ (z) or ζ(z) π z to a compact disk, answering questions from Masser's paper, the latter having been suggested by Pila. As in Masser's original work, the constant C may be effectively computed from certain data associated with the function in question.

Abstract. We consider the question of when an expansion of a topological structure has the property that every open set definable in the expansion is definable in the original structure. This question is related to and inspired by recent work of Dolich, Miller and Steinhorn on the property of having o-minimal open core. We answer the question in a fairly general setting and provide conditions which in practice are often easy to check. We give a further characterisation in the special case of an expansion by a generic predicate.

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Suppose that \Omega is a lattice in the complex plane and let \sigma be the corresponding Weierstrass \sigma -function. Assume that the point \tau associated with \Omega in the standard fundamental domain has imaginary part at most 1.9. Assuming that \Omega has algebraic invariants g_2,g_3 we show that a bound of the form c d^m (\log H)^n holds for the number of algebraic points of height at most H and degree at most d lying on the graph of \sigma . To prove this we apply results by Masser and Besson. What is perhaps surprising is that we are able to establish such a bound for the whole graph, rather than some restriction. We prove a similar result when, instead of g_2,g_3 , the lattice points are algebraic. For this we naturally exclude those (z,\sigma(z)) for which z\in\Omega .

Answering a special case of a question of Chernikov and Simon, we show that any non-dividing formula over a model M in a distal NIP theory is a member of a consistent definable family, definable over M .