Produce a set A such that r ( n ) &gt; 0 for all n ∈ [ 1 , N ], but with | A | ≤ 4 N + 1 .Note that r ( n ) = | { ( a , a ′ ) : a , a ′ ∈ A , n = a + a ′ } | A = { 0 , 1 , 2 } would work with the interval being [ 1 , 4 ]. Then 3 ≤ 17 .A second part of the question shows that one can prove that | A | ≤ N if it satisfies the above conditions. But 3 &gt; 4 = 2. Does this mean that my set A is wrong?

Question

Produce a set   A such that   r  (  n  )  &amp;gt;  0 for all   n  ∈  [  1  ,  N  ], but with       |    A      |    ≤      4    N    +    1  .Note that  r  (  n  )  =      |    {    (    a    ,          a      ′        )    :    a    ,          a      ′        ∈    A    ,    n    =    a    +          a      ′        }    |    A  =  {  0  ,  1  ,  2  } would work with the interval being   [  1  ,  4  ]. Then   3  ≤      17  .A second part of the question shows that one can prove that       |    A      |    ≤      N   if it satisfies the above conditions. But   3  &amp;gt;      4    =  2. Does this mean that my set   A is wrong?

Peutiedw · Accepted Answer

It should be       |    A      |    ≥      N  , since you need at least that many numbers in   A to form enough pairs to produce all the   N numbers.

Produce a set A such that r(n)>0 for all n in [1,N], but with |A|<= sqrt(4N+1). Note that r(n)=∣∣{(a,a′):a,a′ in A,n=a+a′}∣∣

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