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In this paper, we consider the so‐called “Furstenberg set problem” in high dimensions. First, following Wolff's work on the two‐dimensional real case, we provide “reasonable” upper bounds for the problem for   R   or    F  p   . Next we study the “critical” case and improve the “trivial” exponent by    Ω  (  1  /   n  2   )    for    F   p   n   . Our key tool in obtaining this lower bound is a theorem about how things behave when the Loomis–Whitney inequality is nearly sharp, as it helps us to reduce the problem to dimension two.

ON CONFIGURATIONS WHERE THE LOOMIS–WHITNEY INEQUALITY IS NEARLY SHARP AND APPLICATIONS TO THE FURSTENBERG SET PROBLEM