On additive properties of sets defined by the Thue-Morse word

In this paper we study some additive properties of subsets of the set \nats of positive integers: A subset $A$ of \nats is called {\it $k$-summable} (where k\in\ben) if $A$ contains \textstyle \big{\sum_{n\in F}x_n | \emp\neq F\subseteq {1,2,...,k\} \big} for some $k$-term sequence of natural numbers $x_1<x_2 < ... < x_k$. We say A \subseteq \nats is finite FS-big if $A$ is $k$-summable for each positive integer $k$. We say is A \subseteq \nats is infinite FS-big if for each positive integer $k,$ $A$ contains {\sum_{n\in F}x_n | \emp\neq F\subseteq \nats and #F\leq k} for some infinite sequence of natural numbers $x_1<x_2 < ...$. We say A\subseteq \nats is an IP-set if $A$ contains {\sum_{n\in F}x_n | \emp\neq F\subseteq \nats and #F<\infty} for some infinite sequence of natural numbers $x_1<x_2 < ...$. By the Finite Sums Theorem [5], the collection of all IP-sets is partition regular, i.e., if $A$ is an IP-set then for any finite partition of $A$, one cell of the partition is an IP-set. Here we prove that the collection of all finite FS-big sets is also partition regular. Let \TM =011010011001011010... denote the Thue-Morse word fixed by the morphism $0\mapsto 01$ and $1\mapsto 10$. For each factor $u$ of \TM we consider the set \TM\big|_u\subseteq \nats of all occurrences of $u$ in \TM. In this note we characterize the sets \TM\big|_u in terms of the additive properties defined above. Using the Thue-Morse word we show that the collection of all infinite FS-big sets is not partition regular

A New Lower Bound for the Distinct Distance Constant

The reciprocal sum of Zhang sequence is not equal to the Distinct Distance Constant. This note introduces a $B_2$-sequence with larger reciprocal sum, and provides a more precise estimation of the reciprocal sums of Mian-Chowla sequence and Zhang sequence.Comment: 4 pages, 3 ancillary table

Arithmetic-Progression-Weighted Subsequence Sums

Let $G$ be an abelian group, let $S$ be a sequence of terms $s_1,s_2,...,s_{n}\in G$ not all contained in a coset of a proper subgroup of $G$, and let $W$ be a sequence of $n$ consecutive integers. Let $$W\odot S=\{w_1s_1+...+w_ns_n:\;w_i {a term of} W,\, w_i\neq w_j{for} i\neq j\},$$ which is a particular kind of weighted restricted sumset. We show that $|W\odot S|\geq \min\{|G|-1,\,n\}$, that $W\odot S=G$ if $n\geq |G|+1$, and also characterize all sequences $S$ of length $|G|$ with $W\odot S\neq G$. This result then allows us to characterize when a linear equation $$a_1x_1+...+a_rx_r\equiv \alpha\mod n,$$ where $\alpha,a_1,..., a_r\in \Z$ are given, has a solution $(x_1,...,x_r)\in \Z^r$ modulo $n$ with all $x_i$ distinct modulo $n$. As a second simple corollary, we also show that there are maximal length minimal zero-sum sequences over a rank 2 finite abelian group $G\cong C_{n_1}\oplus C_{n_2}$ (where $n_1\mid n_2$ and $n_2\geq 3$) having $k$ distinct terms, for any $k\in [3,\min\{n_1+1,\,\exp(G)\}]$. Indeed, apart from a few simple restrictions, any pattern of multiplicities is realizable for such a maximal length minimal zero-sum sequence

"Weak yet strong'' restrictions of Hindman's Finite Sums Theorem

We present a natural restriction of Hindman’s Finite Sums Theorem that admits a simple combinatorial proof (one that does not also prove the full Finite Sums Theorem) and low computability-theoretic and proof-theoretic upper bounds, yet implies the existence of the Turing Jump, thus realizing the only known lower bound for the full Finite Sums Theorem. This is the first example of this kind. In fact we isolate a rich family of similar restrictions of Hindman’s Theorem with analogous propertie

On a Conjecture of Hamidoune for Subsequence Sums

Let G be an abelian group of order m, let S be a sequence of terms from G with k distinct terms, let m ∧ S denote the set of all elements that are a sum of some m-term subsequence of S, and let |S| be the length of S. We show that if |S| ≥ m + 1, and if the multiplicity of each term of S is at most m − k + 2, then either |m ∧ S| ≥ min{m, |S| − m + k − 1}, or there exists a proper, nontrivial subgroup Ha of index a, such that m ∧ S is a union of Ha-cosets, Ha ⊆ m ∧ S, and all but e terms of S are from the same Ha-coset, where e ≤ min{|S|−m+k−2 |Ha| − 1, a − 2} and |m ∧ S| ≥ (e + 1)|Ha|. This confirms a conjecture of Y. O. Hamidoune