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Sums of positive density subsets of the primes

Kaisa Matomäki (2013)

Acta Arithmetica

We show that if A and B are subsets of the primes with positive relative lower densities α and β, then the lower density of A+B in the natural numbers is at least ( 1 - o ( 1 ) ) α / ( e γ l o g l o g ( 1 / β ) ) , which is asymptotically best possible. This improves results of Ramaré and Ruzsa and of Chipeniuk and Hamel. As in the latter work, the problem is reduced to a similar problem for subsets of * m using techniques of Green and Green-Tao. Concerning this new problem we show that, for any square-free m and any A , B * m of densities α and β, the...

Sums of Powered Characteristic Roots Count Distance-Independent Circular Sets

Zdzisław Skupień (2013)

Discussiones Mathematicae Graph Theory

Significant values of a combinatorial count need not fit the recurrence for the count. Consequently, initial values of the count can much outnumber those for the recurrence. So is the case of the count, Gl(n), of distance-l independent sets on the cycle Cn, studied by Comtet for l ≥ 0 and n ≥ 1 [sic]. We prove that values of Gl(n) are nth power sums of the characteristic roots of the corresponding recurrence unless 2 ≤ n ≤ l. Lucas numbers L(n) are thus generalized since L(n) is the count in question...

Sumsets in quadratic residues

I. D. Shkredov (2014)

Acta Arithmetica

We describe all sets A p which represent the quadratic residues R p 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.

Sumsets of Sidon sets

Imre Z. Ruzsa (1996)

Acta Arithmetica

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 N 1 / 3 is possible. In [1] they show that A + A contains...

Supercongruences for the Almkvist-Zudilin numbers

Tewodros Amdeberhan, Roberto Tauraso (2016)

Acta Arithmetica

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.

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