Elementary And Integral-elementary Functions

By an integral-elementary function we mean any real function that can be obtained from the constants, sin x, e(x), log x, and arcsin x (defined on (-1, 1)) using the basic algebraic operations, composition and integration. The rank of an integral-elementary function f is the depth of the formula defining f. The integral-elementary Functions of rank less than or equal to n are real-analytic and satisfy a common algebraic differential equation P-n(f, f',..., f((k))) = 0 with integer coefficients. We prove that every continuous function f: R --> R can be approximated uniformly by integral-elementary functions of bounded rank. Consequently, there exists an algebraic differential equation with integer coefficients such that its everywhere analytic solutions approximate every continuous function uniformly. This solves a problem posed by L. A. Rubel. Using the same basic functions as above, but allowing only the basic algebraic operations and compositions, we obtain the class of elementary functions. We show that every differentiable function with a derivative not exceeding an iterated exponential can be uniformly approximated by elementary functions of bounded rank. If we include the function arcsin x defined on [-1, 1], then the resulting class of naive-elementary functions will approximate every continuous function uniformly. We also show that every sequence can be uniformly approximated by elementary functions, and that every integer sequence can be represented in the form f(n), where f is naive-elementary

Growth in solvable subgroups of GL_r(Z/pZ)

Let $K=Z/pZ$ and let $A$ be a subset of \GL_r(K) such that is solvable. We reduce the study of the growth of $A$ under the group operation to the nilpotent setting. Specifically we prove that either $A$ grows rapidly (meaning $|A\cdot A\cdot A|\gg |A|^{1+\delta}$), or else there are groups $U_R$ and $S$, with $S/U_R$ nilpotent such that $A_k\cap S$ is large and $U_R\subseteq A_k$, where $k$ is a bounded integer and $A_k = \{x_1 x_2...b x_k : x_i \in A \cup A^{-1} \cup {1}}$. The implied constants depend only on the rank $r$ of $\GL_r(K)$. When combined with recent work by Pyber and Szab\'o, the main result of this paper implies that it is possible to draw the same conclusions without supposing that is solvable.Comment: 46 pages. This version includes revisions recommended by an anonymous referee including, in particular, the statement of a new theorem, Theorem

Sum-free sets in abelian groups

Let A be a subset of an abelian group G. We say that A is sum-free if there do not exist x,y and z in A satisfying x + y = z. We determine, for any G, the cardinality of the largest sum-free subset of G. This equals c(G)|G| where c(G) is a constant depending on G and lying in the interval [2/7,1/2]. We also estimate the number of sum-free subsets of G. It turns out that log_2 of this number is c(G)|G| + o(|G|), which is tight up to the o-term. For certain abelian groups, those whose order is divisible by a small prime of the form 3k + 2, we can obtain an asymptotic for the number of sum-free sets

Distance graphs with finite chromatic number

AbstractThe distance graph G(D) with distance set D={d1, d2, …} has the set Z of integers as vertex set, with two vertices i, j∈Z adjacent if and only if |i−j|∈D. We prove that the chromatic number of G(D) is finite whenever inf{di+1/di}>1 and that every growth speed smaller than this admits a distance set D with infinite-chromatic G(D)