Fraenkel's Partition and Brown's Decomposition

Denote the sequence ([ (n-x') / x ])_{n=1}^\infty by B(x, x'), a so-called Beatty sequence. Fraenkel's Partition Theorem gives necessary and sufficient conditions for B(x, x') and B(y, y') to tile the positive integers, i.e., for B(x, x') \cap B(y, y') = \emptyset and B(x, x') \cup B(y, y') = {1,2, 3, ...}. Fix 0 < x < 1, and let c_k = 1 if k \in B(x, 0), and c_k = 0 otherwise, i.e., c_k=[ (k+1) / x ] - [ k / x]. For a positive integer m let C_m be the binary word c_1c_2c_3... c_m. Brown's Decomposition gives integers q_1, q_2, ..., independent of m and growing at least exponentially, and integers t, z_0, z_1, z_2, ..., z_t (depending on m) such that C_m = C_{q_t}^{z_t}C_{q_{t-1}}^{z_{t-1}} ... C_{q_1}^{z_1}C_{q_0}^{z_0}. In other words, Brown's Decomposition gives a sparse set of initial segments of C_\infty and an explicit decomposition of C_m (for every m) into a product of these initial segments.Comment: 19 page

Sturmian Words and the Permutation that Orders Fractional Parts

A Sturmian word is a map W from the natural numbers into {0,1} for which the set of {0,1}-vectors F_n(W):={(W(i),W(i+1),...,W(i+n-1))^T : i \ge 0} has cardinality exactly n+1 for each positive integer n. Our main result is that the volume of the simplex whose n+1 vertices are the n+1 points in F_n(W) does not depend on W. Our proof of this motivates studying algebraic properties of the permutation $\pi$ (depending on an irrational x and a positive integer n) that orders the fractional parts {1 x}, {2 x}, ..., {n x}, i.e., 0 < {\pi(1) x} < {\pi(2) x} < ... < {\pi(n) x} < 1. We give a formula for the sign of $\pi$, and prove that for every irrational x there are infinitely many n such that the order of $\pi$ (as an element of the symmetric group S_n) is less than n.Comment: 20 pages, 1 figure, Mathematica notebook available from autho

A Discrete Fourier Kernel and Fraenkel's Tiling Conjecture

The set B_{p,r}^q:=\{\floor{nq/p+r} \colon n\in Z \} with integers p, q, r) is a Beatty set with density p/q. We derive a formula for the Fourier transform \hat{B_{p,r}^q}(j):=\sum_{n=1}^p e^{-2 \pi i j \floor{nq/p+r} / q}. A. S. Fraenkel conjectured that there is essentially one way to partition the integers into m>2 Beatty sets with distinct densities. We conjecture a generalization of this, and use Fourier methods to prove several special cases of our generalized conjecture.Comment: 24 pages, 6 figures (now with minor revisions and clarifications

The supremum of autoconvolutions, with applications to additive number theory

We adapt a number-theoretic technique of Yu to prove a purely analytic theorem: if f(x) is in L^1 and L^2, is nonnegative, and is supported on an interval of length I, then the supremum of the convolution f*f is at least 0.631 \| f \|_1^2 / I. This improves the previous bound of 0.591389 \| f \|_1^2 / I. Consequently, we improve the known bounds on several related number-theoretic problems. For a subset A of {1,2, ..., n}, let g be the maximum multiplicity of any element of the multiset {a+b: a,b in A}. Our main corollary is the inequality gn>0.631|A|^2, which holds uniformly for all g, n, and A.Comment: 17 pages. to appear in IJ

On sequences without geometric progressions

An improved upper bound is obtained for the density of sequences of positive integers that contain no k-term geometric progression.Comment: 4 pages; minor correctio

A problem of Rankin on sets without geometric progressions

A geometric progression of length $k$ and integer ratio is a set of numbers of the form $\{a,ar,\dots,ar^{k-1}\}$ for some positive real number $a$ and integer $r\geq 2$. For each integer $k \geq 3$, a greedy algorithm is used to construct a strictly decreasing sequence $(a_i)_{i=1}^{\infty}$ of positive real numbers with $a_1 = 1$ such that the set $$G^{(k)} = \bigcup_{i=1}^{\infty} \left(a_{2i} , a_{2i-1} \right]$$ contains no geometric progression of length $k$ and integer ratio. Moreover, $G^{(k)}$ is a maximal subset of $(0,1]$ that contains no geometric progression of length $k$ and integer ratio. It is also proved that there is a strictly increasing sequence $(A_i)_{i=1}^{\infty}$ of positive integers with $A_1 = 1$ such that $a_i = 1/A_i$ for all $i = 1,2,3,\ldots$. The set $G^{(k)}$ gives a new lower bound for the maximum cardinality of a subset of the set of integers $\{1,2,\dots,n\}$ that contains no geometric progression of length $k$ and integer ratio.Comment: 15 page