Sumsets avoiding squarefree integers
We describe all sets which represent the quadratic residues in the sense that R = A + A or R = A ⨣ A. Also, we consider the case of an approximate equality R ≈ A + A and R ≈ A ⨣ A and prove that A is then close to a perfect difference set.
1. Introduction. A Sidon set is a set A of integers with the property that all the sums a+b, a,b∈ A, a≤b are distinct. A Sidon set A⊂ [1,N] can have as many as (1+o(1))√N elements, hence N/2 sums. The distribution of these sums is far from arbitrary. Erdős, Sárközy and T. Sós [1,2] established several properties of these sumsets. Among other things, in [2] they prove that A + A cannot contain an interval longer than C√N, and give an example that is possible. In [1] they show that A + A contains...
We prove a conjecture on supercongruences for sequences that have come to be known as the Almkvist-Zudilin numbers. Some other (naturally) related family of sequences will be considered in a similar vain.
On étudie, dans cet article, les propriétés combinatoires de mots engendrés à l’aide de -automates déterministes dénombrables de degré borné, ou de manière équivalente, engendrés par des substitutions de longueur constante uniformément bornées sur un alphabet dénombrable. En particulier, on montre que la complexité de tels mots est au plus polynomiale et que, sur plusieurs exemples, elle est au plus de l’ordre de grandeur de .