We improve upon the traditional error term in the truncated Perron formula for the logarithm of an L-function. All our constants are explicit.

In this paper we prove a commutative algebraic extension of a generalized Skolem–Mahler– Lech theorem. Let $A$ be a finitely generated commutative $K$–algebra over a field of characteristic 0, and let $\sigma$ be a $K$–algebra automorphism of $A$. Given ideals $I$ and $J$ of $A$, we show that the set $S$ of integers $m$ such that ${{\sigma }^{m}}(I)\,\supseteq \,J$ is a finite union of complete doubly infinite arithmetic progressions in $m$, up to the addition of a finite set. Alternatively, this result states that for an affine scheme $X$ of finite type over $K$, an automorphism $\sigma \,\in \,\text{Au}{{\text{t}}_{k}}(X)$, and $Y$ and $Z$ any two closed subschemes of $X$, the set of integers $m$ with ${{\sigma }^{m}}(Z)\,\subseteq \,Y$ is as above. We present examples showing that this result may fail to hold if the affine scheme $X$ is not of finite type, or if $X$ is of finite type but the field $K$ has positive characteristic.

Benford's law (to base $B$) for an infinite sequence $\{x_k: k \ge 1\}$ of positive quantities $x_k$ is the assertion that $\{ \log_B x_k : k \ge 1\}$ is uniformly distributed $(\bmod\ 1)$. The $3x+1$ function $T(n)$ is given by $T(n)=(3n+1)/{2}$ if $n$ is odd, and $T(n)= n/2$ if $n$ is even. This paper studies the initial iterates $x_k= T^{(k)}(x_0)$ for $1 \le k \le N$ of the $3x+1$ function, where $N$ is fixed. It shows that for most initial values $x_0$, such sequences approximately satisfy Benford's law, in the sense that the discrepancy of the finite sequence $\{\log_B x_k: 1 \le k \le N \}$ is small.

This paper studies the problem of characterizing the simplest aperiodic discrete point sets, using invariants based on topological dynamics. A Delone set of finite type is a Delone set X such that X-X is locally finite. Such sets are characterized by their patch-counting function N_X (T) of radius T being finite for all T. We formulate conjectures relating slow growth of the patch-counting function N_X (T) to the set X having a non-trivial translation symmetry.

A Delone set X of finite type is repetitive if there is a function M_X(T) such that every closed ball of radius M_X(T) + T contains a complete copy of each kind of patch of radius T that occurs in X. This is equivalent to the minimality of an associated topological dynamical system with \mathbb{R}^n-action. There is a lower bound for M_X(T) in terms of N_X(T), namely M_X(T) \ge c( N_X (T))^{1/n} for some positive constant c depending on the Delone set constants r,R, but there is no general upper bound for M_X(T) purely in terms of N_X(T). The complexity of a repetitive Delone set X is measured by the growth rate of its repetitivity function M_X (T). For example, the function M_X (T) is bounded if and only if X is a periodic crystal. A set X is linearly repetitive if M_X (T) = O(T) as T \to \infty and is densely repetitive if M_X (T) = O(N_X(T))^{1/n} as T \to \infty. We show that linearly repetitive sets and densely repetitive sets have strict uniform patch frequencies, i.e. the associated topological dynamical system is strictly ergodic. It follows that such sets are diffractive, in the sense of having a well-defined diffraction measure. In the reverse direction, we construct a repetitive Delone set X in \mathbb{R}^n which has M_X (T) = O(T (\log T)^{2/n} (\log\log\log T )^{4/n}), but does not have uniform patch frequencies. Aperiodic linearly repetitive sets have many claims to be the simplest class of aperiodic sets and we propose considering them as a notion of ‘perfectly ordered quasicrystals’.

This paper characterizes when a Delone set $X$ in ${{\mathbb{R}}^{n}}$ is an ideal crystal in terms of restrictions on the number of its local patches of a given size or on the heterogeneity of their distribution. For a Delone set $X$, let ${{N}_{X}}\left( T \right)$ count the number of translation-inequivalent patches of radius $T$ in $X$ and let ${{M}_{X}}\left( T \right)$ be the minimum radius such that every closed ball of radius ${{M}_{X}}\left( T \right)$ contains the center of a patch of every one of these kinds. We show that for each of these functions there is a “gap in the spectrum” of possible growth rates between being bounded and having linear growth, and that having sufficiently slow linear growth is equivalent to $X$ being an ideal crystal.

Explicitly, for ${{N}_{X}}\left( T \right)$, if $R$ is the covering radius of $X$ then either ${{N}_{X}}\left( T \right)$ is bounded or ${{N}_{X}}\left( T \right)\,\ge \,T/2R$ for all $T\,>\,0$. The constant $1/2R$ in this bound is best possible in all dimensions.

For ${{M}_{X}}\left( T \right)$, either ${{M}_{X}}\left( T \right)$ is bounded or ${{M}_{X}}\left( T \right)\ge T/3$ for all $T\,>\,0$. Examples show that the constant 1/3 in this bound cannot be replaced by any number exceeding 1/2. We also show that every aperiodic Delone set $X$ has ${{M}_{X}}\left( T \right)\,\ge \,c\left( n \right)T$ for all $T\,>\,0$, for a certain constant $c\left( n \right)$ which depends on the dimension $n$ of $X$ and is $>\,1/3$ when $n\,>\,1$.

Linear mod one transformations are those maps of the unit interval given by $f_{\beta,\alpha}(x)=\beta x+\alpha$ (mod 1), with $\beta>1$ and $0\le\alpha<1$. The lap-counting function is $L_{\beta,\alpha}(z)=\sum_{n=1}^{\infty} L_{n}z^{n}$, where $L_{n}$ essentially counts the number of monotonic pieces of the $n$th iterate $f_{\beta,\alpha}^{n}$. Part I showed that the function $L_{\beta,\alpha}(z)$ is meromorphic in the unit disk $\vert z\vert <1$ and analytic in $\vert z\vert<1/\beta$, and part II showed that the singularities of $L_{\beta,\alpha}(z)$ on the circle $\vert z\vert=1/\beta$ are contained in the set $\{(1/\beta)\exp (2\pi il/N_{\beta,\alpha}):0\le l/N_{\beta,\alpha}\}$, where $N_{\beta,\alpha}$ is the period of the ergodic part of a Markov chain associated to $f_{\beta,\alpha}$. This paper proves that the set of singularities on $\vert z\vert=1/\beta$ is identical to the set $\{(1/\beta)\exp (2\pi il/N_{\beta,\alpha}):0\le l/N_{\beta,\alpha}\}$. Part II showed that $N_{\beta,\alpha}=1$ for $\beta> 2$, and this paper determines $N_{\beta,\alpha}$ in the remaining cases where $1<\beta\le 2$.

Linear mod one transformations are those maps of the unit interval given by $f_{\beta,\alpha}(x)=\beta x+\alpha$ (mod 1), with $\beta>1$ and $0\le\alpha<1$. The lap-counting function is $L_{\beta,\alpha}(z)=\sum_{n=1}^{\infty} L_{n}z^{n}$, where $L_{n}$ essentially counts the number of monotonic pieces of the $n$th iterate $f_{\beta,\alpha}^{n}$. Part I showed that the function $L_{\beta,\alpha}(z)$ is meromorphic on the unit disk $|z|<1$ and analytic on $|z|<1/\beta$. This paper shows that the singularities of $L_{\beta,\alpha}(z)$ on the circle $|z|=1/\beta$ are contained in the set $\{(1/\beta)\exp (2\pi il/N):0\le l\le N-1\}$, for some integer $N\ge 1$. Here $N$ can be taken to be the period $N_{\beta,\alpha}$ of a certain Markov chain $\Sigma_{\beta,\alpha}$ which encodes information about generalized lap numbers $L_{n}(i,j)$ of $f_{\beta,\alpha}$, where $L_{n}(i,j)$ counts monotonic pieces of $f_{\beta,\alpha}^{n}$ whose image is $[f^{i}(0),f^{j}(1^{-}))$. We show that $N_{\beta,\alpha}=1$ whenever $\beta>2$. Finally, we give the criterion that $N_{\beta,\alpha}=1$ if and only if for all $n\ge 1$ the map $f_{\beta,\alpha}^{n}$ is ergodic with respect to the maximal entropy measure of $f_{\beta,\alpha}$.

The 3x+1 map T and the shift map S are defined by T(x) = (3x + 1)/2 for x odd, T(x) = x/2 for x even, while S(x) = (x − 1)/2 for x odd, S(x) = x/2 for x even. The 3x + 1 conjugacy map Φ on the 2-adic integers Z2 conjugates S to T, i.e., Φ o S o Φ-1 = T. The map Φ mod 2n induces a permutation Φn on Z/2nZ. We study the cycle structure of Φn. In particular we show that it has order 2n − 4 for n ≥ 6. We also count 1-cycles of Φn for n up to 1000; the results suggest that Φ has exactly two odd fixed points. The results generalize to the ax + b map, where ab is odd.

Linear mod one transformations are the maps of the unit interval given by fβα(x) = βx + α (mod 1), with β > 1 and 0 ≤ α < 1. The lap-counting function is the function where the lap number Ln essentially counts the number of monotonic pieces of the nth iterate . We derive an explicit factorization formula for Lβα(z) which directly shows that Lβα(z) is a function meromorphic in the open unit disk {z: |z| < 1} and analytic in the open disk {z: |z| < 1/β}, with a simple pole at z = 1/β.Comparison with a known formula for the Artin—Mazur—Ruelle zeta function ζβ,α(z) of fβα shows that Lβα(z) and ζβ,α(z) have identical sets of singularities in the disk {z: |z| < 1}. We derive two more factorization formulae for Lβ,α(z) valid for certain parameter ranges of (β, α). When 1 < α + β ≤ 2, there is sometimes a ‘renormalization’ structure of such maps present, which has previously been studied in connection with simplified models for the Lorenz attractor. In the case that fβα is non-trivially renormalizable, we obtain a factorization formula for Lβα(z). For (β, α) in a region contained in 2 < α + β ≤ 3 we obtain a factorization formula which relates Lβα(z) to a ‘rescaled’ lap-counting function from the region 1 < α + β ≤ 2. The various factorizations exhibit certain singularities of Lβα(z) on the circle |z| = 1/β. These singularities are related to topological dynamical properties of fβ,α. In parts II and III we show that these comprise the complete set of such singularities on the circle |z| = 1/β.

Szekeres defined a continuous analogue of the additive ordinary continued fraction expansion, which iterates a map T on a domain which can be identified with the unit square [0, 1]2. Associated to it are continuous analogues of the Lagrange and Markoff spectrum. Our main result is that these are identical with the usual Lagrange and Markoff spectra, respectively; thus providing an alternative characterization of them.

Szekeres also described a multi-dimensional analogue of T, which iterates a map Td on a higherdimensional domain; he proposed using it to bound d-dimensional Diophantine approximation constants. We formulate several open problems concerning the Diophantine approximation properties of the map Td.

The β-transformation ƒβ(x) = βx(mod 1), for β > 1, has a symbolic dynamics generalizing radix expansions to an integer base. Two important invariants of ƒβ are the (Artin-Mazur) zeta function

where Pk counts the number of fixed points of , and the lap-counting function where Lk counts the number of monotonic pieces of the kth iterate . For β-transformations these functions are related by ζβ(z) = (1 − z)Lβ(z). The function ζβ(z) is meromorphic in the unit disk, is holomorphic in {z: |z| < 1/β}, has a simple pole at z = 1/β, and has no other singularities with |z| = 1/β. Let M(β) denote the minimum modulus of any pole of ζβ(z) in |z| < 1 other than z = 1/β, and set M(β) = 1 if no other pole exists with |z| < 1. Then Pk = βk + O((M(β)−1+ε)k) for any ε > 0. This paper shows that M(β) is a continuous function, that ( for all β, and that An asymptotic formula is derived for M(β) as β → 1+, which implies that M(β) < 1 for all β in an interval (1, 1 + c0). The set is shown to have properties analogous to the set of Pisot numbers. It is closed, totally disconnected, has smallest element ≥ 1 + C0 and contains infinitely many β falling in each interval [n, n + 1) for n ∈ ℤ+. All known members of are algebraic integers which are either Pisot or Salem numbers.

A lattice polytope is a polytope in whose vertices are all in . The volume of a lattice polytope P containing exactly k ≥ 1 points in d in its interior is bounded above by . Any lattice polytope in of volume V can after an integral unimodular transformation be contained in a lattice cube having side length at most n˙n ! V. Thus the number of equivalence classes under integer unimodular transformations of lattice poly topes of bounded volume is finite. If S is any simplex of maximum volume inside a closed bounded convex body K in having nonempty interior, then K⊆ ( n + 2)S — (n+ l)s where mS denotes a nomothetic copy of S with scale factor m, and s is the centroid of S.