A density threshold of 1 - O((log n)/n) forces measurable sets in R^d to contain arbitrarily large similar copies of any n-point configuration, matching the upper bound up to a logarithmic factor.
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Near-optimal density theorems for large dilates of large point configurations
A density threshold of 1 - O((log n)/n) forces measurable sets in R^d to contain arbitrarily large similar copies of any n-point configuration, matching the upper bound up to a logarithmic factor.