In 1962, Stanislaw Ulam defined a sequence of integers starting with 1 and 2, and continuing by always choosing the next largest integer that is a sum of two distinct smaller elements of the sequence in a unique way. This sequence, called the ``Ulam numbers,'' has been puzzling ever since, with its deeply recursive definition making it difficult to analyse, as though the sequence were in some way random. Then, in 2015, Stefan Steinerberger noted a remarkable ``hidden signal'' in the sequence--a unexpectedly large value of the Fourier transform of the sequence's indicator function occuring at some irrational number $\alpha$, and that this gives rise to a non-uniform distribution of the Ulam numbers modulo $\frac{2\pi}{\alpha}$. In this document, we provide some theorems in the direction of validating Steinerberger's observations, both of the large Fourier coefficient and of the non-uniform distribution. These results in fact apply to a wider class of sets that all have fewer than expected solutions to the equation $x + y = z$, including sum-free sets, the Ulam numbers, more general 1-additive sets, and others. We also state stronger versions of our theorems which, if certain phenomena that we observe by computer calculations were to be proven true, would follow from applications of a circle method technique.