Higher Moments and Phase-Riesz Products for the Lonely Runner Conjecture

For a set V of n distinct positive integer speeds, writeML(V)=max_{t in T} min_{v in V} ||tv||_T. Bedert proved the uniformasymptotic lower bound ML(V) >= 1/(2n) + 1/n^{5/3+o(1)}. We refine the large-additive-dimension branch of his argument and prove ML(V) >= 1/(2n) + 1/n^{13/8+o(1)}. The new ingredient is a Riesz product with independent phases. Thecovering inequality produces a nonnegative function on phase space withvery small mean. A level-two Fourier inequality then forces thequadratic signed-sum coefficients of the covering function to have smalll^2-energy. Bedert's arithmetic-progression intersection estimateconverts this into an l^1-bound for the quadratic Riesz chaos with anadditional factor equal to the covering excess. The remaining Rieszlevels begin cubically. Balancing the resulting large-dimension estimatewith Bedert's short-model estimate yields the exponent 13/8.