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The purpose of this work is twofold. First, we aim to extend for $0<s<1$ theresults of one of the authors about equilibrium measures in the real axis inexternal fields created by point-mass charges for the case of logarithmicpotentials ($s=0$).  Our second motivation comes from the work of the other two authors on Riesz$s$-equilibrium problems on finitely many intervals on the real line in thepresence of external fields. They have shown that when the signed equilibriummeasure has concave positive part on every interval, then the $s$-equilibriumsupport is also a union of finitely many intervals, with one of them at mostincluded in each of the initial intervals. As the positive part of the signedequilibrium for point-mass external fields is not necessarily concave, theinvestigation of the corresponding $s$-equilibrium support is of comparativeinterest. Moreover, we provide simple examples of compactly supportedequilibrium measures in external fields $Q$ not satisfying the usualrequirements about the growth at infinity, that is, $\displaystyle\lim_{|x|\rightarrow \infty}\,Q(x) = \infty\,.$  Our main tools are signed equilibrium measures and iterated balayagealgorithm in the context of Riesz $s$-equilibrium problems on the real line. Asthese techniques are not mainstream work in the field and can be applied inother contexts we highlight their use here.

On point-mass Riesz external fields on the real axis