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We consider problems where many, somewhat redundant, hypotheses are testedand we are interested in reporting the most precise rejections, with falsediscovery rate (FDR) control. For example, a common goal in genetics is toidentify DNA variants that carry distinct information on a trait of interest.However, strong local dependencies between nearby variants make it challengingto distinguish which of the many correlated features most directly influencethe phenotype. A common solution is then to identify sets of variants thatcover the truly important ones. Depending on the signal strengths, it ispossible to resolve the individual variant contributions with more or lessprecision. Assuring FDR control on the reported findings with these adaptivesearches is, however, often impossible. To design a multiple comparisonprocedure that allows for an adaptive choice of resolution with FDR control, weleverage e-values and linear programming. We adapt this approach to problemswhere knockoffs and group knockoffs have been successfully applied to testconditional independence hypotheses. We demonstrate its efficacy by analyzingdata from the UK Biobank.

Catch me if you can: Signal localization with knockoff e-values