GPT 6 Astra Ultra attempt on Math-5544
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Contents
Erdős Problem #275
Statement and attribution
If $r$ residue classes $a_i\pmod{n_i}$ cover $2^r$ consecutive integers, must they cover all integers? The moduli may repeat. Yes, and $2^r$ is sharp. This known theorem was proved by Selfridge and by Crittenden--Vanden Eynden; Balister, Bollobás, Morris, Sahasrabudhe and Tiba gave a short algebraic proof. We reconstruct the algebraic mechanism using a Vandermonde matrix. No novelty is claimed.Work: sparse exponential sums
For integer $m$, define \[ F(m)=\prod_{i=1}^r\left(1-\exp\left(\frac{2\pi\mathrm i(m-a_i)}{n_i}\right)\right). \] A factor vanishes exactly when $m\equiv a_i\pmod{n_i}$. Therefore $F(m)=0$ exactly at the covered integers. Expanding the product and combining equal frequencies yields \[ F(m)=\sum_{j=1}^t c_j\lambda_j^m, \qquad t\le2^r, \] where the $\lambda_j$ are distinct nonzero complex numbers. Zero coefficients can be discarded; if none remain, the conclusion is already proved.Suppose the covered interval starts at $b$. Its first $t$ integers give \[ \sum_{j=1}^t(c_j\lambda_j^b)\lambda_j^\ell=0 \qquad(0\le\ell<t). \] The coefficient matrix has determinant $\prod_{i<j}(\lambda_j-\lambda_i)\ne0$. Hence every $c_j\lambda_j^b$ is zero, so every $c_j$ is zero. It follows that $F(m)=0$ for every integer $m$, including negative integers. This proves the theorem.
Sharpness and a consequence
For $1\le i\le r$, take $2^{i-1}\pmod{2^i}$. Each integer $1\le m<2^r$ has a unique exponent $0\le v_2(m)<r$ and lies in the class with $i=v_2(m)+1$. The system covers $2^r-1$ consecutive integers but misses every multiple of $2^r$, so the threshold cannot be decreased.The same theorem shows that in a minimal covering system with $r$ classes, each modulus is at most $2^{r-1}$. Remove a class of modulus $d$. The remaining $r-1$ classes still cover the $d-1$ integers strictly between consecutive members of the removed class, but do not cover all integers by minimality. Therefore $d-1<2^{r-1}$, or $d\le2^{r-1}$.
Verification and outcome
Repeated moduli or coincident subset frequencies cause no problem because frequencies are combined before applying the matrix argument. A modulus one makes $F$ identically zero and is also harmless. All ingredients of the proof and sharpness example have been supplied.SOLVED.