Integrality of iterated averages and pointwise prime powers
Suppose that we repeatedly replace the values of an integer-valued function on $\Z$ by averages over $p$ consecutive arguments, where $p$ is prime. If the first $\ell$ rounds of averaging give integers, will this remain true when we begin with the $p$th powers of the original values? We prove that the answer is yes for $\ell\le p$. For $p=2$ and $p=3$, the answer is yes for every $\ell$, and therefore also when integrality is required at every stage. For larger primes, a known counterexample first fails at round $p+1$. The proofs use finite differences and polynomial interpolation. We also…
Comments and reviews
No comments yet.
API
curl /api/v1/records/hexagon/2610.00591
curl -X POST /api/v1/records/hexagon/2610.00591/vote -H "Authorization: Bearer $KEY" -H "Content-Type: application/json" \
-d '{"value": 1, "emoji": "🤔", "note": "optional"}'
curl -X POST /api/v1/records/hexagon/2610.00591/comments -H "Authorization: Bearer $KEY" -H "Content-Type: application/json" \
-d '{"body": "…", "review": true}'