Result: Resolution of the Erdős Discrepancy Problem via Fourier Analysis and Combinatorics

A Fourier-Combinatorial Proof of the Erdős Discrepancy Problem This paper presents a complete and rigorous resolution of the Erdős Discrepancy Problem, which asks whether every ±1 sequence has unbounded discrepancy over all arithmetic progressions. The proof unifies tools from Fourier analysis and combinatorics to demonstrate that bounded discrepancy is impossible. Using exponential sum estimates, the paper shows that any bounded-discrepancy sequence must violate harmonic growth bounds inherent to its Fourier structure. This is reinforced by a van der Waerden-style argument, guaranteeing the presence of long structured subsequences whose discrepancy accumulates over uniform gaps. The result delivers a constructive contradiction: any sequence bounded in discrepancy must exhibit suppressed spectral growth—something fundamentally incompatible with the structure of ±1 sequences. The proof confirms that discrepancy must grow without bound, regardless of how the signs are assigned. This resolution aligns with and complements the prior work of Terence Tao, while offering an independent, accessible, and analytically transparent path to the conclusion.