A Appendix: A Short History of Universality Thus it appears that universality is a generic phenomenon in analysis. K.-G. Grosse-Erdmann We conclude with a brief historical overview on the phenomenon of universal- ity. We refer to Grosse-Erdmann [109] for a detailed and interesting survey on more or less all known types of universalities and an approach to unify them all. The first universal object appearing in the mathematical literature was dis- covered by Fekete in 1914/15 (see [287]); he proved that there exists a real ∞ n power series n=1 anx with the property that for any continuous function f(x) on the interval [−1, 1] satisfying f(0) = 0 there is a sequence of positive integers mk such that mk n anx −→ f(x) uniformly on [−1, 1]. k→∞ n=1 The proof relies essentially on Weierstrass’ approximation theorem which states that every continuous function on a compact interval is the limit of a uniformly convergent sequence of polynomials (see [367]). We illustrate Fekete’s theorem and its proof by a p-adic version. The p-adic numbers were introduced by Hensel [131] in 1897. Given a prime number p, we define the p-adic absolute value | . |p on Z by p−ν(α;p) if α =0 , |α| = p 0 otherwise; here ν(α; p) is the exponent of p in the prime factorization of α ∈ Z.Itisan easy task to extend |·|p from Z to Q.Thep-adic absolute values satisfies the strong triangle inequality: for any α, β ∈ Q, |α + β|p ≤ max{|α|p, |β|p} 286 Appendix in contrast to the standard absolute value. The p-adic absolute value is non- archimedean (i.e., sup{|n|p : n ∈ N} is bounded). The integers that are p-adically close to zero are precisely the ones that are highly divisible by p. The set of p-adic numbers is the completion of Q with respect to the p-adic Q absolute value, and is denoted by p.Thep-adic absolute value extends to a Q Q non-archimedean absolute value on p,and p becomes a complete, locally compact, totally disconnected Hausdorff space. Similarly to the completion R Q Q of with respect to the standard (archimedean) absolute value, p is a field, and its elements have a p-adically convergent series representation: for any ∈ Q α p, there exist integers ν and ak such that k α = akp with 0 ≤ ak <p. k≥ν(α;p) The set of p-adic numbers α with |α|p ≤ 1 forms a ring, the ring Zp of p-adic integers. Another construction of p-adic numbers uses the representation of k Zp as a projective limit of the ring of residue classes mod p . It is customary to write |·|∞ for the standard absolute value on Q, Q∞ for R,andthen call R the completion of Q at the infinite prime p = ∞. Two absolute values are called equivalent if they induce the same topology. Ostrowski (see [279, Sect. II.3]) showed that every non-trivial absolute value on Q is equivalent to one of the absolute values |·|p,wherep is a prime number or p = ∞.Thus, completion of Q with respect to its non-equivalent absolute values yields a Q Q family of complete, locally compact topological fields p which contain , one for each place p ≤∞: Q → Q Q Q Q R " 2, 3, 5, ..., and ∞ = . p-adic analysis is quite different from real analysis. A non-archimedean absolute value induces a curious (ultrametric) topology. In p-adic analysis, the role of the intervals in R are played by the balls ν Z { ∈ Q | − | ≤ −ν } a + p p := α p : α a p p , ∈ Q ∈ Z where a p and ν . These balls are clopen sets, i.e., they are closed and open. It follows that each point inside a ball is center of the ball, and that any two balls are either disjoint or contained one in another. Q Q Q The algebraic closure p of p has infinite degree over p, but is not complete, and so it is not the right field for doing analysis. However, the C Q completion p of p is algebraically closed. If 2 2 n = α0 + α1p + α2p + ··· and k = β0 + β1p + β2p + ··· are the p-adic expansions of the integers n and k, then one can show that n α α ≡ 0 1 ··· mod p. k β0 β1 A Short History of Universality 287 In view of the binomic inversion formula this leads for any continuous functions f : Zp → Cp to the representation n n f(n)= a (f) , k k k=0 ∈ N where n and k k a (f):= (−1)k−j f(j). k j j=0 It is remarkable that this yields a series representation, the so-called Mahler series of f.Fork ∈ N0, we define the finite difference operator " by k k "kf(x)= (−1)k−j f(x + j). j j=0 k Then ak = " f(0) for k ∈ N. Mahler [233] proved Theorem A.1. Let f : Zp → Cp be a continuous function. Then ak → 0 as k →∞,and n x ak −→ f(x) uniformly on Zp. k n→∞ k=0 A proof can be found in Robert [311, Sect. 4.2.4]. Notice that this theorem does not hold over R. In particular, Mahler’s Theorem A.1 implies a p-adic version of Weierstrass’ approximation theorem: for any continuous function f : Zp → Cp, there exists a sequence of polynomials fn with coefficients in Cp that converges uniformly to f. Further applications of Mahler’s theorem are the construc- tion of p-adic analogues of classical functions like the exponential function, the logarithm or the Gamma-function; for details we refer again to Robert’s monograph [311]. Now, it is not too surprising that an analogue of Fekete’s universality theorem holds in the p-adic case too. Steuding [341] obtained ∞ n Theorem A.2. There exists a p-adic power series n=1 anx with the prop- ν erty that for any ball a+p Zp,wherea, ν ∈ Z, 0 <a<p, and any continuous ν function f, defined on a+p Zp and satisfying f(a)=0, there exists a sequence of positive integers mk such that mk n ν anx −→ f(x) uniformly on a + p Zp. k→∞ n=1 We follow Luh [227] in his proof of a version of Fekete’s theorem. 288 Appendix Proof. Let {Qn} be a sequence of all polynomials with integral coefficients. We construct a sequence of polynomials {Pn} as follows: let P0 = Q0 and assume that P0,...,Pn−1 are known. Denote by dn the degree of Pn−1.Now let φn be a continuous function on Zp such that n−1 −dn−1 φn(x)= Qn(x) − Pk(x) x k=0 −n for x ∈ a+p Zp. In view of Mahler’s Theorem A.1 there exists a polynomial fn so that max |f (x) − φ (x)| ≤ p−dn . −n n n p x∈a+p Zp dn+1 Setting Pn(x)=fn(x)x , the sequence {Pn} is constructed. We note n max P (x) − Q (x) = max |(f (x) − φ (x))xdn+1| −n k n −n n n p x∈a+p Zp x∈a+p Zp k=0 p ≤ p−dn . By construction, distinct Pn have no powers in common. Thus we can rearrange formally the polynomial series into a power series: ∞ ∞ n anx := Pn(x). n=1 n=0 ν Again by Mahler’s Theorem A.1, for any continuous function f on a + p Zp there exists a sequence of positive integers nk, tending to infinity with k,such that max |f(x) − Q (x)| ≤ p−nk . −n nk p x∈a+p k Zp ν −nk For sufficiently large k the ball a + p Zp is contained in a + p Zp.Inview of the above estimates we obtain dnk +1 nk n max f(x) − anx = max f(x) − Pn(x) −n −n x∈a+p k Zp x∈a+p k Zp n=1 n=0 p p ⎧ ⎫ ⎨ k ⎬ n ≤ max |f(x) − Q (x)| , Q (x) − P (x) −n nk p nk n x∈a+p k Zp ⎩ ⎭ n=0 p − − ≤ max{p nk ,p dnk }, →∞ which tends to zero as k . Thus, putting mk = dnk +1, the assertion of the theorem follows. A Short History of Universality 289 In the years after Fekete’s discovery many universal objects were found. For instance, Birkhoff [23] proved in 1929 the existence of an entire function f(z) with the property that to any given entire function g(z) there exists a sequence of complex numbers an such that f(z + an) −→ g(z) uniformly on compacta in C. n→∞ The proof relies in the main part on Runge’s approximation theorem. This type of universality is very similar to the one of the Riemann zeta-function and other Dirichlet series. Birkhoff’s theorem states the existence of an entire function with wild behaviour near infinity. Luh [228] constructed holomorphic monsters, that are holomorphic functions with an extraordinary wild bound- ary behaviour in arbitrary simply connected open sets. More precisely: let G be a proper open subset of C with simply connected components. Then there exists a function f holomorphic on G such that for every boundary point z of G,everycompactsubsetK with connected complement and every continuous function g on K which is holomorphic in the interior of K, there exist linear transformations τn(z)=anz + bn with τn(K) ⊂Gand dist(τn(K),z) → 0 as n →∞for which f(τn(z)) −→ g(z) uniformly on K; n→∞ in addition, each derivative of f and each antiderivative of f of arbitrary order has the boundary behaviour described above.