Sumsets being squares
نویسندگان
چکیده
Alon, Angel, Benjamini and Lubetzky recently studied an old problem of Euler on sumsets for which all elements of A + B are integer squares. Improving their result we prove: 1. There exists a set A of 3 positive integers and a corresponding set B ⊂ [0, N ] with |B| ≫ (logN)15/17, such that all elements of A+B are perfect squares. 2. There exists a set A of 3 integers and a corresponding set B ⊂ [0, N ] with |B| ≫ (logN)9/11, such that all elements of the sets A, B and A+B are perfect squares. The proofs make use of suitably constructed elliptic curves of high rank.
منابع مشابه
The additive structure of the squares inside rings
When defining the amount of additive structure on a set it is often convenient to consider certain sumsets; Calculating the cardinality of these sumsets can elucidate the set’s underlying structure. We begin by investigating finite sets of perfect squares and associated sumsets. We reveal how arithmetic progressions efficiently reduce the cardinality of sumsets and provide estimates for the min...
متن کاملSquares in Sumsets
A finite set A of integers is square-sum-free if there is no subset of A sums up to a square. In 1986, Erdős posed the problem of determining the largest cardinality of a square-sum-free subset of {1, . . . , n}. Answering this question, we show that this maximum cardinality is of order n1/3+o(1).
متن کاملA Quantitative Result on Diophantine Approximation for Intersective Polynomials
In this short note, we closely follow the approach of Green and Tao to extend the best known bound for recurrence modulo 1 from squares to the largest possible class of polynomials. The paper concludes with a brief discussion of a consequence of this result for polynomial structures in sumsets and limitations of the method.
متن کاملSquares in Sumsets
A finite set A of integers is square-sum-free if there is no subset of A sums up to a square. In 1986, Erd˝ os posed the problem of determining the largest cardinality of a square-sum-free subset of {1,. .. , n}. Answering this question, we show that this maximum cardinality is of order n 1/3+o(1) .
متن کاملLattice Points on Circles, Squares in Arithmetic Progressions and Sumsets of Squares
Let σ(k) denote the maximum of the number of squares in a+b, . . . , a+kb as we vary over positive integers a and b. Erdős conjectured that σ(k) = o(k) which Szemerédi [30] elegantly proved as follows: If there are more than δk squares amongst the integers a+b, . . . , a+kb (where k is sufficiently large) then there exists four indices 1 ≤ i1 < i2 < i3 < i4 ≤ k in arithmetic progression such th...
متن کامل