Existence of a Sidon set for the distinct distance constant

Abstract We highlight a certain compactness of Sidon sets and $$B_2[g]$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>B</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mrow> <mml:mo>[</mml:mo> <mml:mi>g</mml:mi> <mml:mo>]</mml:mo> </mml:mrow> </mml:mrow> </mml:math> -sets and provide several applications. Notably, we prove the existence of such sets that maximize certain functions. In particular, we show the existence of a Sidon set whose reciprocal sum is equal to the distinct distance constant. We also improve the best known bounds for this constant.