When an Entropy Constant Falls to Minus Infinity: Two Errors in Tao's Theorem 1.4

This technical note identifies two mathematical errors in Theorem 1.4 of Terence Tao's Sum-difference exponents for boundedly many slopes, and rational complexity (arXiv:2511.15135v1). First, the variational constant \(c_\alpha\) defined in equation (1.9) satisfies \(c_0=-\infty\), contradicting the asserted bound \(0<c_\alpha<\infty\). An explicit family of admissible smooth probability densities establishes this counterexample. Second, the entropy asymptotics on pages 17–18 yield the coefficient \(2B-A\), rather than the \(A-B\) appearing in equation (5.1). This algebraic discrepancy invalidates the stated derivation. The note provides rigorous calculations, identifies the precise locations of both errors, and discusses their mathematical implications. These findings concern Theorem 1.4 as written and do not disprove the separate Theorem 1.3. Keywords: Differential entropy, sum-difference exponents, variational bounds, mathematical correction, Kakeya problem.