MathDB

A nonempty set

A

of real numbers is called a

B_3

-set if the conditions

a_1, a_2, a_3, a_4, a_5, a_6 \in A

and a_1 \plus{} a_2 \plus{} a_3 \equal{} a_4 \plus{} a_5 \plus{} a_6 imply that the sequences

(a_1, a_2, a_3)

and

(a_4, a_5, a_6)

are identical up to a permutation. Let

A = \{a_0 = 0 < a_1 < a_2 < \cdots \}

B = \{b_0 = 0 < b_1 < b_2 < \cdots \}

be infinite sequences of real numbers with D(A) \equal{} D(B), where, for a set

X

of real numbers,

D(X)

denotes the difference set \{|x\minus{}y|\mid x, y \in X \}. Prove that if

A

is a

B_3

-set, then A \equal{} B.

Problem Statement