Let (x_1, x_2, x_3, …) be a sequence with x_i ∈ +1, -1 for every positive integer i, and let C be an integer. Then there exist positive integers k and d such that |∑_i=1^k x_i · d| > C. Equivalently, for every such sequence, sup_k, d ≥ 1 |∑_i=1^k x_i · d| = ∞, the supremum being taken over all pairs of positive integers k and d.

Erdős discrepancy problem

About 94 years old · traced to 1932

Let $(x_1, x_2, x_3, \ldots)$ be a sequence with $x_i \in \{+1, -1\}$ for every positive integer $i$ , and let $C$ be an integer. Then there exist positive integers $k$ and $d$ such that

\left|\sum_{i=1}^{k} x_{i \cdot d}\right| > C.

Equivalently, for every such sequence,

\sup_{k, d \geq 1} \left|\sum_{i=1}^{k} x_{i \cdot d}\right| = \infty,

the supremum being taken over all pairs of positive integers $k$ and $d$ .

Equivalent formulations 2Other wordings

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

Erdős discrepancy problem
In mathematics, a sign sequence, or ±1–sequence or bipolar sequence, is a sequence of numbers, each of which is either 1 or −1. One example is the sequence.
source: Wikipedia
Erdős Problem #67 — The Erdős Discrepancy Problem
For every function $f\colon\mathbb{N}\to\{-1,+1\}\subset\mathbb{R}$ and every real number $C>0$ , there exist natural numbers $d,m\geq 1$ such that
$\left|\sum_{1\leq k\leq m} f(kd)\right|>C.$
source: The Formal Conjectures Authors, Formal Conjectures (2025); independently edited from the pinned source file.

References

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Wikipedia

Additional references

Wikipedia, Sign sequence, the article this problem comes from.

discrepancy unsolved-problem

InErdős Problems

Posted 2 months ago · edited 3 weeks ago

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Erdős discrepancy problem

Erdős Problem #67 — The Erdős Discrepancy Problem

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