  We consider problems where many, somewhat redundant, hypotheses are tested
and we are interested in reporting the most precise rejections, with false
discovery rate (FDR) control. This is the case, for example, when researchers
are interested both in individual hypotheses as well as group hypotheses
corresponding to intersections of sets of the original hypotheses, at several
resolution levels. A concrete application is in genome-wide association
studies, where, depending on the signal strengths, it might be possible to
resolve the influence of individual genetic variants on a phenotype with
greater or lower precision. To adapt to the unknown signal strength, analyses
are conducted at multiple resolutions and researchers are most interested in
the more precise discoveries. Assuring FDR control on the reported findings
with these adaptive searches is, however, often impossible. To design a
multiple comparison procedure that allows for an adaptive choice of resolution
with FDR control, we leverage e-values and linear programming. We adapt this
approach to problems where knockoffs and group knockoffs have been successfully
applied to test conditional independence hypotheses. We demonstrate its
efficacy by analyzing data from the UK Biobank.
